Bivariate models and regularizing priors

Elizabeth King

Kevin Middleton

Bivariate models

\[y = \beta_0 + \beta_1 x\]

Need to explicitly include the intercept
$x$ is continuous: “linear regression”
$x$ is categorical: $t$-test or ANOVA
- $\beta_0 == 0$

Simulate data

$\beta_0 = 2$
$\beta_1 = 6.7$

set.seed(4534759)
D <- tibble(x = runif(20, 0, 10),
            y = 2 + 6.7 * x + rnorm(20, 0, 3))

ggplot(D, aes(x, y)) +
  geom_point(size = 4)

Simulate data

Model statements

$y$ follows a normal distribution with a mean and standard deviation:

\[y \sim Normal(\mu, \sigma)\]

The mean is a linear function of an intercept ($a$) and a slope ($b$):

\[\mu = \beta_0 + \beta_1 \cdot x\]

$\beta_0$ and $\beta_1$ are arbitrary names.

With more predictors, more meaningful names are useful

Prior prediction: Reasonable values for $\beta_0$, $\beta_1$, and $\sigma$

Think about the slope first
Intercept depends on the slope, so let it vary more
- Intercept is often not “interesting”
- Byproduct of the other parameters
Don’t need to think about $\sigma$ for prior prediction (usually)

Normal distributions

$\sim 95\%$ of points fall within $\pm 2$ standard deviations of the mean

Center parameter estimates on 0 (unless you have a reason otherwise a priori)
- “Slope not different from zero”
- “No difference in means”
- “No difference between groups”

Regularizing priors

Bayesian priors are conservative
- All values not equally likely (true for maximum likelihood)
Less likely to find an effect by chance
Lessens the influence of extreme or unusual values automatically

Prior prediction

Intercept probably $0 \pm 40$
Slope probably $0 \pm 10$

set.seed(3457522)

sim_model <- function(ii) {
  b0 <- rnorm(1, 0, 20)
  b1 <- rnorm(1, 0, 5)
  tibble(rep = ii,
         x = seq(0:10),
         y = b0 + b1 * x)
}

sims <- purrr::map(.x = 1:50,
            .f = sim_model) |> 
  list_rbind()
print(sims, n = 200)

Prior prediction

# A tibble: 550 × 3
      rep     x        y
    <int> <int>    <dbl>
  1     1     1  -39.2  
  2     1     2  -50.3  
  3     1     3  -61.5  
  4     1     4  -72.6  
  5     1     5  -83.7  
  6     1     6  -94.8  
  7     1     7 -106.   
  8     1     8 -117.   
  9     1     9 -128.   
 10     1    10 -139.   
 11     1    11 -151.   
 12     2     1   11.6  
 13     2     2   23.3  
 14     2     3   35.0  
 15     2     4   46.7  
 16     2     5   58.4  
 17     2     6   70.1  
 18     2     7   81.8  
 19     2     8   93.5  
 20     2     9  105.   
 21     2    10  117.   
 22     2    11  129.   
 23     3     1  -24.6  
 24     3     2  -26.7  
 25     3     3  -28.8  
 26     3     4  -30.9  
 27     3     5  -33.0  
 28     3     6  -35.1  
 29     3     7  -37.2  
 30     3     8  -39.3  
 31     3     9  -41.4  
 32     3    10  -43.5  
 33     3    11  -45.6  
 34     4     1  -31.4  
 35     4     2  -29.9  
 36     4     3  -28.4  
 37     4     4  -26.9  
 38     4     5  -25.4  
 39     4     6  -23.9  
 40     4     7  -22.4  
 41     4     8  -20.9  
 42     4     9  -19.4  
 43     4    10  -17.9  
 44     4    11  -16.4  
 45     5     1   10.1  
 46     5     2    9.47 
 47     5     3    8.85 
 48     5     4    8.23 
 49     5     5    7.61 
 50     5     6    6.99 
 51     5     7    6.37 
 52     5     8    5.75 
 53     5     9    5.13 
 54     5    10    4.50 
 55     5    11    3.88 
 56     6     1   28.3  
 57     6     2   33.8  
 58     6     3   39.3  
 59     6     4   44.8  
 60     6     5   50.3  
 61     6     6   55.8  
 62     6     7   61.3  
 63     6     8   66.8  
 64     6     9   72.3  
 65     6    10   77.8  
 66     6    11   83.3  
 67     7     1  -22.9  
 68     7     2  -28.8  
 69     7     3  -34.8  
 70     7     4  -40.8  
 71     7     5  -46.8  
 72     7     6  -52.8  
 73     7     7  -58.7  
 74     7     8  -64.7  
 75     7     9  -70.7  
 76     7    10  -76.7  
 77     7    11  -82.6  
 78     8     1   10.4  
 79     8     2    9.99 
 80     8     3    9.54 
 81     8     4    9.09 
 82     8     5    8.64 
 83     8     6    8.19 
 84     8     7    7.74 
 85     8     8    7.29 
 86     8     9    6.84 
 87     8    10    6.39 
 88     8    11    5.94 
 89     9     1    2.47 
 90     9     2   -2.38 
 91     9     3   -7.22 
 92     9     4  -12.1  
 93     9     5  -16.9  
 94     9     6  -21.8  
 95     9     7  -26.6  
 96     9     8  -31.4  
 97     9     9  -36.3  
 98     9    10  -41.1  
 99     9    11  -46.0  
100    10     1   24.3  
101    10     2   23.9  
102    10     3   23.5  
103    10     4   23.1  
104    10     5   22.6  
105    10     6   22.2  
106    10     7   21.8  
107    10     8   21.4  
108    10     9   21.0  
109    10    10   20.6  
110    10    11   20.2  
111    11     1    6.99 
112    11     2   10.6  
113    11     3   14.1  
114    11     4   17.7  
115    11     5   21.2  
116    11     6   24.8  
117    11     7   28.4  
118    11     8   31.9  
119    11     9   35.5  
120    11    10   39.1  
121    11    11   42.6  
122    12     1   46.9  
123    12     2   50.2  
124    12     3   53.5  
125    12     4   56.8  
126    12     5   60.1  
127    12     6   63.4  
128    12     7   66.7  
129    12     8   70.0  
130    12     9   73.3  
131    12    10   76.6  
132    12    11   79.9  
133    13     1  -15.6  
134    13     2  -11.5  
135    13     3   -7.46 
136    13     4   -3.38 
137    13     5    0.696
138    13     6    4.77 
139    13     7    8.85 
140    13     8   12.9  
141    13     9   17.0  
142    13    10   21.1  
143    13    11   25.2  
144    14     1  -38.5  
145    14     2  -47.1  
146    14     3  -55.7  
147    14     4  -64.2  
148    14     5  -72.8  
149    14     6  -81.4  
150    14     7  -90.0  
151    14     8  -98.5  
152    14     9 -107.   
153    14    10 -116.   
154    14    11 -124.   
155    15     1   23.3  
156    15     2   22.8  
157    15     3   22.3  
158    15     4   21.7  
159    15     5   21.2  
160    15     6   20.6  
161    15     7   20.1  
162    15     8   19.6  
163    15     9   19.0  
164    15    10   18.5  
165    15    11   17.9  
166    16     1   19.8  
167    16     2   21.8  
168    16     3   23.7  
169    16     4   25.7  
170    16     5   27.6  
171    16     6   29.6  
172    16     7   31.5  
173    16     8   33.5  
174    16     9   35.4  
175    16    10   37.4  
176    16    11   39.3  
177    17     1  -24.3  
178    17     2  -16.5  
179    17     3   -8.75 
180    17     4   -0.991
181    17     5    6.77 
182    17     6   14.5  
183    17     7   22.3  
184    17     8   30.1  
185    17     9   37.8  
186    17    10   45.6  
187    17    11   53.3  
188    18     1   16.3  
189    18     2   22.0  
190    18     3   27.6  
191    18     4   33.2  
192    18     5   38.9  
193    18     6   44.5  
194    18     7   50.1  
195    18     8   55.8  
196    18     9   61.4  
197    18    10   67.0  
198    18    11   72.7  
199    19     1   -5.43 
200    19     2   -7.69 
# … with 350 more rows

Plotting the prior predictions

Are the predicted outcomes on the right scale?

ggplot(sims, aes(x, y, group = rep)) +
  geom_line(alpha = 0.5)

Plotting the prior predictions

Fitting the model

fm <- ulam(
  alist(
    y ~ normal(mu, sigma),
    mu <- b0 + b1 * x,
    b0 ~ dnorm(0, 20),
    b1 ~ dnorm(0, 5),
    sigma ~ dhalfnorm(0, 5)
  ),
  data = D,
  chains = 4,
  iter = 5e3
)

Fitting the model

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Examining the stan code

stancode(fm)

data{
    vector[20] y;
    vector[20] x;
}
parameters{
    real b0;
    real b1;
    real<lower=0> sigma;
}
model{
    vector[20] mu;
    sigma ~ normal( 0 , 5 );
    b1 ~ normal( 0 , 5 );
    b0 ~ normal( 0 , 20 );
    for ( i in 1:20 ) {
        mu[i] = b0 + b1 * x[i];
    }
    y ~ normal( mu , sigma );
}

Inspecting the chains: `traceplot`

traceplot(fm)

Inspecting the chains: Rank histogram

trankplot(fm)

Inspecting the output

precis(fm)

          mean        sd      5.5%    94.5%    n_eff    Rhat4
b0    1.800517 2.0608328 -1.416041 5.049027 2142.901 1.000487
b1    6.783442 0.3078417  6.294372 7.278068 2239.678 1.000694
sigma 3.420959 0.6215384  2.592630 4.512934 3261.233 1.001175

Compare to lm()

fm_lm <- lm(y ~ x, data = D)
coef(fm_lm)

(Intercept)           x 
   1.651613    6.807963

confint(fm_lm, level = 0.89)

                5.5 %   94.5 %
(Intercept) -1.558829 4.862056
x            6.328467 7.287459

Posteriors

post <- extract.samples(fm) |> 
  as.data.frame()
post

                b0      b1   sigma
1      1.00020e+00 6.83154 2.76616
2      6.08167e+00 6.06179 3.49124
3     -1.24904e+00 7.02523 3.31848
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1637   3.53908e+00 6.53438 3.55049
1638   1.50953e+00 6.87401 2.53142
1639   2.86882e+00 6.57237 2.70251
1640   1.27048e+00 6.74772 3.02771
1641   1.46084e+00 6.96009 3.52426
1642   1.50684e+00 6.91384 3.86073
1643   3.27922e+00 6.57293 2.62934
1644   2.38906e+00 6.72925 3.10977
1645   9.92875e-01 7.14775 3.91973
1646   3.82625e+00 6.51765 3.76365
1647   4.10556e+00 6.35593 3.95509
1648  -6.59893e-01 7.36035 2.92954
1649  -7.49110e-01 7.34439 5.28018
1650   6.90840e-01 6.95131 2.97200
1651  -2.31573e+00 7.44774 4.37722
1652   1.75435e+00 6.64834 3.29852
1653   2.14747e+00 6.47667 3.09567
1654   2.75995e+00 6.82742 2.71417
1655   1.75225e+00 6.52838 4.54839
1656   3.55279e+00 6.49985 4.09845
1657   3.01884e+00 6.56818 3.25518
1658  -1.56832e+00 7.02926 3.55152
1659   3.00578e+00 6.63136 3.42778
1660   7.22438e-01 6.94419 3.23563
1661   3.93989e+00 6.26505 4.64872
1662   4.97174e+00 6.25942 4.06345
1663   2.41454e+00 6.73617 5.01001
1664   5.92724e+00 6.25982 3.45243
1665  -1.07594e+00 7.09144 3.34320
1666  -1.76967e-01 7.07896 2.66385
1667   7.07128e-01 6.97027 2.92256
1668   1.76300e+00 6.84003 2.52477
1669   9.77950e-01 7.03889 3.57371
1670   3.65051e+00 6.55612 2.99714
1671   6.97995e+00 6.12811 4.82015
1672   1.28689e+00 7.06617 3.36183
1673   5.35347e+00 6.22984 2.74827
1674  -8.27838e-01 7.03209 3.40916
1675  -2.71292e+00 7.42710 2.47214
1676   5.74517e+00 6.21980 3.14012
1677   1.41610e+00 6.71421 2.61077
1678   3.58739e+00 6.46037 2.75231
1679   2.52375e+00 6.68559 3.36585
1680   3.33617e+00 6.60462 3.21010
1681  -2.40868e+00 7.60338 3.79592
1682   5.10087e+00 6.50521 4.19088
1683  -9.19543e-01 7.01957 2.95665
1684   2.65373e+00 6.74584 3.18639
1685   4.54639e+00 6.33878 2.76894
1686   3.66759e+00 6.38409 3.13792
1687   4.42474e+00 6.59544 3.80396
1688   1.10035e+00 6.92594 3.72228
1689   2.17392e+00 6.91009 3.38486
1690   4.32631e-01 6.93328 2.81464
1691  -1.80445e-01 7.07082 2.60429
1692   1.23908e+00 6.90395 4.67190
1693   2.80319e+00 6.53034 4.48048
1694   2.74764e+00 6.48298 3.52320
1695   1.81760e+00 6.85699 2.59463
1696   8.73156e-01 6.83948 2.91701
1697   3.19507e+00 6.43590 3.36134
1698   1.12339e+00 6.66041 2.96206
1699   4.75886e+00 6.44260 3.01997
1700   1.99743e+00 6.90314 3.21946
1701   1.52812e+00 6.66028 3.53408
1702   2.22216e+00 6.62132 3.14593
1703  -7.56837e-01 6.83569 3.30067
1704   6.67338e-01 7.05173 4.37143
1705   9.23205e+00 5.86884 5.79453
1706  -2.05615e-02 7.10810 3.69901
1707   4.27872e-01 7.04233 3.54408
1708   3.84520e+00 6.47654 3.81535
1709  -4.02224e-01 6.84982 3.68972
1710   4.59424e+00 6.39658 4.01767
1711   1.33293e+00 6.81059 3.48274
1712  -2.91378e+00 7.44287 3.21186
1713   3.03808e+00 6.62977 3.06302
1714   2.64203e+00 6.75413 2.50160
1715  -2.05675e-01 6.82632 3.16970
1716   2.20713e+00 6.71512 2.62244
1717   4.17268e+00 6.40352 3.16703
1718   1.93393e+00 6.74374 4.20346
1719   4.59859e+00 6.39286 2.96889
1720   3.05906e+00 6.88669 4.28426
1721   3.12906e+00 6.52944 3.72277
1722   4.13384e+00 6.33518 2.56713
1723   2.07196e+00 6.78892 3.27641
1724   1.49824e+00 6.87336 2.77674
1725   9.61358e-01 7.01961 4.77469
1726   2.64828e+00 6.46219 3.04960
1727   1.51751e+00 6.74990 2.86389
1728   2.10231e-01 6.88449 3.99975
1729   2.47096e+00 6.83011 3.47911
1730   1.23001e+00 6.80808 3.92097
1731   3.39459e+00 6.70503 3.42782
1732  -8.85027e-01 7.14850 3.69954
1733   1.72138e+00 6.95420 3.28499
1734   1.09152e+00 6.72485 3.31386
1735  -1.71532e+00 7.31713 4.41239
1736   1.76481e+00 6.87446 3.05133
1737   3.63085e+00 6.66952 3.34182
1738   1.37512e-01 7.02746 3.73617
1739   1.43059e+00 6.81783 3.43427
1740  -1.21735e-01 6.96413 5.72087
1741   8.21086e-01 6.98950 3.14194
1742  -1.06461e+00 7.00860 4.16860
1743   1.57553e-01 7.07756 3.54418
1744  -1.81084e+00 7.37997 4.08305
1745  -5.31968e+00 7.87690 3.62284
1746   7.33138e+00 6.01721 3.99122
1747   3.68164e+00 6.26136 2.91624
1748  -1.00350e+00 7.11826 3.85895
1749   2.36294e+00 6.60249 3.31364
1750   8.21898e-02 7.11130 3.26224
1751  -1.43564e+00 7.00147 3.91582
1752   4.02986e+00 6.39541 2.82181
1753   1.75767e+00 6.65759 3.73430
1754   2.27719e+00 6.62721 2.86436
1755   2.78589e+00 6.72538 2.99964
1756   3.31751e+00 6.56960 3.41750
1757   9.06097e-01 6.89301 2.33212
1758   3.27059e+00 6.60002 3.33071
1759   3.53799e+00 6.59943 3.07292
1760   1.27343e+00 6.89265 3.41122
1761  -8.86240e-01 7.17935 3.18718
1762   5.33957e-01 7.01767 2.79417
1763  -1.69132e-01 7.05869 3.50310
1764  -8.35125e-01 7.29738 2.75755
1765   4.27363e+00 6.49493 4.35490
1766  -5.17083e-01 7.15146 3.02671
1767   1.34488e+00 6.66630 3.46043
1768   1.91963e+00 6.76858 2.99341
1769   3.81381e+00 6.49995 2.88055
1770   7.42867e+00 5.91957 3.66139
1771   1.93472e+00 6.81757 3.43750
1772   1.15270e+00 6.74606 2.27500
1773   9.40987e-01 6.66332 3.51220
1774   3.24548e+00 6.57393 3.29411
1775   3.34693e+00 6.54320 5.18739
1776   4.98210e+00 6.27344 2.89261
1777   6.75024e-01 6.94130 3.21432
1778   9.74091e-01 6.91713 3.33249
1779   9.88658e-01 6.95034 4.26230
1780   3.77576e+00 6.47115 3.01922
1781   1.90320e+00 6.76670 3.77725
1782   2.53834e+00 6.88389 2.88164
1783   4.13400e+00 6.53585 2.79069
1784   3.04366e+00 6.63513 2.90544
1785   4.93071e+00 6.23015 3.26610
1786   1.94482e+00 6.89927 2.48129
1787   9.36593e-01 6.90396 3.25416
1788   2.07610e+00 6.80485 3.43589
1789  -1.30288e+00 7.05246 3.55413
1790   1.56902e+00 6.73017 3.21882
1791   9.89534e-01 6.59176 4.23132
1792   3.78089e-01 6.93682 3.49667
1793   2.27553e+00 6.71079 3.03476
1794   9.22911e-01 7.10255 4.78243
1795   1.13290e+00 6.90443 3.22337
1796   1.02921e+00 6.97199 2.60372
1797   3.30507e+00 6.48007 3.61705
1798   1.13692e+00 7.11481 3.37807
1799   3.95088e+00 6.38925 3.15405
1800   1.30356e-01 7.00281 3.33268
1801   5.65816e+00 6.21039 3.88555
1802   2.98345e+00 6.77281 3.29724
1803   4.81733e+00 6.39264 3.39756
1804   4.16766e+00 6.30009 2.93178
1805   2.00735e+00 6.84552 2.59713
1806   2.30374e+00 6.65106 3.12522
1807   1.56604e+00 7.00646 4.77093
1808   6.25668e-01 6.89566 3.64795
1809   1.74333e+00 6.78199 2.83963
1810   3.85954e+00 6.64167 3.55582
1811   4.68499e-01 6.91447 2.73044
1812   2.04530e+00 6.77123 3.44290
1813   3.22639e+00 6.47895 3.09976
1814   1.30661e+00 7.01485 3.55087
1815   3.18480e-01 6.96830 3.23515
1816   4.73154e+00 6.17613 4.48909
1817   4.02421e+00 6.41554 3.39415
1818   1.46686e+00 6.85783 4.05432
1819   1.34034e+00 6.63495 4.58121
1820   1.08660e+00 6.88089 3.66377
1821   2.04541e+00 6.76204 2.88661
1822   3.26639e+00 6.53913 2.92643
1823  -1.29823e+00 7.18582 2.81482
1824   1.20819e-01 7.07060 3.41069
1825   5.02487e+00 6.20416 3.41756
1826  -4.39028e-02 6.96261 2.95809
1827   2.64348e+00 6.74500 3.34922
1828   4.00092e+00 6.46933 2.74287
1829   4.00024e+00 6.48768 3.69828
1830   3.03500e+00 6.60758 2.67816
1831  -3.81754e-01 7.04029 2.94077
1832   1.91627e+00 6.42594 4.17005
1833   2.29594e+00 6.83757 2.73365
1834  -1.05044e+00 7.21861 4.76912
1835   2.47118e+00 6.50776 3.31136
1836  -1.19577e+00 7.29012 2.93723
1837   3.98638e+00 6.47301 3.06103
1838  -4.98427e-01 7.25996 3.19920
1839   1.82153e+00 6.83156 3.48950
1840   2.72462e+00 6.68525 3.55107
1841  -8.00626e-01 7.03863 3.19081
1842   2.71362e+00 6.61176 3.61494
1843   2.91899e+00 6.61597 4.51320
1844   4.79579e-01 7.00422 4.17730
1845   7.64504e-01 6.99586 3.23506
1846   2.69721e+00 6.57813 3.72569
1847   4.61753e+00 6.31661 3.88069
1848   2.33126e+00 6.66817 3.05158
1849   2.36598e+00 6.54300 3.65764
1850   2.54127e+00 6.74710 3.69930
1851   1.98903e+00 6.90068 2.99749
1852   5.73314e+00 6.19486 4.07214
1853   2.69756e+00 6.52090 2.93838
1854   2.55825e+00 6.74394 3.60463
1855   1.27946e+00 6.72509 3.50780
1856   4.64247e+00 6.37263 2.99961
1857   3.16124e+00 6.52889 4.46813
1858   3.28987e+00 6.58947 2.67743
1859  -1.17728e+00 7.18344 3.52570
1860  -8.72768e-02 7.14135 2.80074
1861   2.31707e+00 6.88633 3.16932
1862   1.78740e+00 6.67879 2.81469
1863   4.63174e+00 6.34725 2.95840
1864   1.29343e-01 6.94109 3.47173
1865   1.23292e+00 6.65709 3.88904
1866   2.56032e+00 6.68286 4.32263
1867   2.71651e+00 6.65722 2.99363
1868   3.50506e+00 6.60310 3.77974
1869  -1.16768e-01 7.00687 3.18022
1870   9.14316e-01 6.97181 4.16949
1871  -1.63217e+00 7.20992 3.69571
1872   3.01894e+00 6.44032 3.39507
1873   1.58256e+00 6.90758 3.13234
1874   1.01505e+00 6.92269 3.98668
1875   2.10665e+00 6.47679 2.80396
1876   8.80611e-01 6.88267 3.94282
1877   2.15841e+00 6.78435 3.06676
1878  -1.54802e+00 7.43734 3.57940
1879   1.90168e+00 6.78174 3.55178
1880   1.04034e+00 6.84905 2.90376
1881   1.61325e-01 7.10421 3.29460
1882   4.24477e+00 6.37771 3.11528
1883   3.37759e+00 6.66581 3.49514
1884  -7.13470e-01 7.09774 3.39356
1885   2.60245e+00 6.88472 2.84808
1886  -6.65911e-01 6.99041 3.82543
1887   1.39979e+00 6.76163 2.80842
1888   2.24093e+00 6.95594 3.47917
1889   4.27463e+00 6.52514 4.80191
1890   4.18995e+00 6.51991 3.21536
1891   3.23697e+00 6.61300 3.23326
1892   2.18923e+00 6.73092 3.21376
1893   1.07378e+00 6.85405 2.91465
1894   3.12761e-01 7.10102 3.57202
1895  -4.26449e-01 7.04789 2.75431
1896   3.38187e+00 6.62701 3.03615
1897   5.23060e-01 7.18202 3.89285
1898   1.04316e+00 6.85903 2.81939
1899   9.25662e-01 6.96227 3.24307
1900   1.33915e+00 6.86431 2.71325
1901  -1.40346e+00 7.13711 3.31641
1902   2.38627e+00 6.72250 3.16970
1903  -1.68039e+00 7.43212 3.75985
1904   1.47374e+00 6.79085 2.33083
1905  -1.61885e+00 7.26969 4.03353
1906  -3.59966e+00 7.26647 3.44921
1907   2.93154e+00 6.73655 2.78689
1908   4.95380e+00 6.21986 3.41414
1909   1.55584e+00 6.69859 3.05800
1910   9.19762e-01 6.91779 2.83499
1911  -8.85708e-01 7.21568 3.58526
1912   2.94799e+00 6.55556 3.27151
1913  -8.40815e-01 7.19991 3.60813
1914   3.35809e+00 6.47109 3.56406
1915   2.06381e+00 6.55261 2.98933
1916   2.25792e+00 6.82098 2.83838
1917   5.51034e-01 6.93035 2.91028
1918   4.71988e+00 6.35714 3.06596
1919   3.97879e+00 6.63998 3.56924
1920   7.86849e-01 6.94894 3.19804
1921   6.30961e-01 6.80811 3.26843
1922   4.46352e+00 6.37841 3.36215
1923  -1.31820e+00 7.01271 3.48494
1924  -1.35948e+00 7.14559 3.48304
1925   2.86132e+00 6.54348 3.56298
1926  -8.49505e-01 6.95407 4.01050
1927   3.50739e+00 6.55150 3.35979
1928   2.54677e+00 6.49235 4.71625
1929   3.21537e-01 7.03587 2.93295
1930   2.33432e+00 6.64528 3.12286
1931   3.07113e+00 6.45459 4.16969
1932  -2.78085e-01 6.90921 3.84200
1933   4.96639e+00 6.41287 3.79558
1934  -6.68404e-01 7.01169 3.80099
1935   3.66520e+00 6.23445 4.50556
1936   3.10942e+00 6.66638 3.59999
1937  -1.58902e-01 7.15222 3.14802
1938   3.81407e+00 6.67539 3.83725
1939  -5.64762e-01 6.92248 2.73993
1940   2.46823e+00 6.84132 3.24732
1941   2.38125e+00 6.74529 3.09728
1942   2.14181e+00 6.65179 3.48089
1943  -3.16122e-01 6.88985 3.91933
1944   7.60785e-01 6.91652 3.38613
1945   2.22061e+00 6.69062 3.91189
1946   2.90506e+00 6.24715 3.17933
1947   3.09456e+00 6.86765 4.61692
1948   4.14306e+00 6.56658 3.55949
1949   1.28773e+00 6.89693 3.63493
1950   3.25395e+00 6.79812 3.10188
1951   2.56637e+00 6.73979 3.31447
1952   3.29789e+00 6.47230 3.46205
1953  -1.78511e+00 7.35190 4.18999
1954   3.82161e+00 6.62701 4.24133
1955   2.76769e+00 6.83599 2.68878
1956  -1.40570e+00 7.20152 4.18351
1957   2.79933e+00 6.66749 2.89367
1958  -2.11908e+00 7.30649 3.45213
1959   6.03258e+00 6.19799 3.95173
1960   1.39197e+00 6.86269 2.70037
1961   1.92436e+00 6.68267 3.35672
1962   1.08145e+00 6.93728 3.74909
1963   1.56246e+00 6.68193 3.70843
1964  -1.25565e+00 7.05262 3.26143
1965   3.13722e+00 6.64493 3.60113
1966  -1.54178e+00 7.32566 3.11524
1967   3.46875e+00 6.49874 3.72366
1968   6.29212e+00 6.24838 3.51072
1969   1.37942e+00 6.92647 3.09654
1970   1.34522e+00 6.85897 3.06700
1971   1.51298e+00 6.79109 2.60021
1972   2.56421e+00 6.65774 3.67258
1973  -1.66416e+00 7.49508 3.74750
1974   1.15991e+00 6.75331 3.16605
1975   1.86916e+00 6.76748 2.41841
1976   5.49143e+00 6.10947 3.33866
1977   3.79987e+00 6.50314 2.90484
1978   1.16809e+00 6.81445 3.40602
1979   3.91715e+00 6.48791 3.08976
1980   1.77130e+00 6.80570 2.92520
1981  -5.84956e-01 7.02840 3.17129
1982   3.63901e+00 6.36098 3.40929
1983   1.04357e+00 6.96789 3.01473
1984   6.19666e-01 6.94679 2.97560
1985  -6.96907e-01 7.06406 3.21720
1986   1.85602e+00 6.64483 3.61188
1987   9.01131e-01 6.98004 4.20886
1988   4.79381e+00 6.16476 4.01829
1989   2.36079e+00 6.79132 3.97232
1990   1.26728e+00 6.74971 3.81481
1991   6.62970e+00 6.02691 3.91010
1992   3.36653e+00 6.56910 4.39152
1993   3.69107e+00 6.50507 3.45658
1994   5.55596e-04 7.04133 2.70393
1995  -2.12050e+00 7.24883 3.70388
1996   2.76868e+00 6.53383 3.39434
1997   3.78063e+00 6.53530 4.47200
1998   2.27959e+00 6.63051 2.40657
1999   4.58451e+00 6.29631 4.66147
2000   2.01028e+00 6.91846 3.53220
2001   2.13457e+00 6.69272 3.39652
2002   1.21480e+00 6.86252 3.68814
2003   9.37278e-01 7.08400 4.47777
2004   5.83344e+00 6.15086 3.07548
2005   2.13116e+00 6.75227 2.46436
2006   1.96922e+00 6.77257 2.69418
2007   3.97736e+00 6.45405 2.75658
2008   8.17174e-02 7.15606 2.97316
2009  -2.00060e-01 7.19483 3.95572
2010   3.83644e+00 6.48891 3.39124
2011   3.82366e+00 6.43939 3.46130
2012   2.65587e+00 6.76403 3.71462
2013   4.07392e-01 7.14348 3.22900
2014   2.23472e+00 6.65821 2.89195
2015  -1.02117e+00 7.06510 3.61064
2016   1.06859e+00 6.87016 2.99538
2017   2.44390e+00 6.70572 2.91379
2018   1.05506e+00 6.88518 4.43357
2019   2.40310e+00 6.65532 3.47518
2020  -9.22263e-02 7.01766 3.49556
2021   1.10400e+00 7.02536 3.85513
2022   2.32486e+00 6.74080 3.49131
2023   2.79934e+00 6.39461 5.93130
2024  -2.66795e+00 7.28242 3.13327
2025  -2.03452e+00 7.21035 3.20425
2026   1.72814e+00 6.71187 2.68886
2027   7.49788e+00 6.09081 4.10324
2028   1.79583e+00 6.80766 2.75337
2029   2.89489e+00 6.25355 3.22538
2030   9.24577e-01 6.66603 4.33679
2031   2.77793e+00 6.63432 3.89243
2032  -2.04711e+00 7.13585 2.93055
2033   4.45393e+00 6.41304 3.42599
2034   2.30174e+00 6.69242 3.30730
2035   2.80411e+00 6.56693 3.42409
2036   4.41698e+00 6.44295 3.86923
2037   3.24098e+00 6.42462 3.58410
2038   1.55730e+00 6.75348 3.60207
2039   2.52217e-01 7.01899 3.78366
2040   1.75464e-01 6.93953 3.14530
2041  -1.68035e-01 6.92069 3.01624
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9607  -1.91881e+00 7.27466 3.11752
9608   3.31031e-01 6.84391 3.25101
9609   2.05849e+00 6.84946 3.17590
9610   1.18384e+00 6.84291 3.53687
9611   3.77919e+00 6.46965 3.06655
9612   1.02592e+00 6.89886 3.11711
9613   2.73725e+00 6.88321 3.18451
9614   1.38338e+00 6.87458 2.50129
9615   3.07959e+00 6.56601 2.50640
9616   3.93175e+00 6.42626 3.21263
9617   1.58268e+00 6.80645 2.75043
9618   5.74045e+00 6.32365 2.88573
9619   1.58570e+00 6.73346 3.08260
9620   4.14226e+00 6.60595 2.51327
9621   2.25832e+00 6.67248 4.12482
9622   3.75152e+00 6.69659 4.31273
9623   2.85846e-01 6.83716 3.26614
9624   2.42418e-01 7.13419 3.97806
9625  -1.61437e+00 7.28199 3.93919
9626   2.72498e-01 6.92820 3.07076
9627   2.76309e+00 6.54920 2.97406
9628  -1.29321e+00 7.21223 3.79499
9629   1.20005e+00 6.92496 2.72578
9630  -9.81827e-02 7.08088 2.67620
9631   1.67904e+00 6.64416 3.23704
9632   9.89175e-01 6.92207 2.63461
9633   3.93254e+00 6.59521 2.96495
9634   1.57552e+00 6.84621 4.48191
9635  -1.92273e+00 7.34166 4.30270
9636   1.50544e+00 6.72843 2.92402
9637   1.43784e+00 6.90955 3.31272
9638   5.65184e-01 7.15887 4.30432
9639   1.74555e+00 6.70703 3.54754
9640   3.04685e+00 6.73904 2.98183
9641   3.53139e+00 6.73740 3.88277
9642   5.12217e+00 6.05192 3.63697
9643  -6.00092e+00 7.88539 4.21593
9644   2.02665e+00 6.66108 3.28382
9645   8.76107e-01 6.70683 3.01453
9646   1.27992e+00 6.82696 2.82031
9647   1.37726e+00 6.86856 4.76738
9648   4.58661e+00 6.36728 3.26405
9649   2.48142e-01 7.05290 3.75075
9650   2.41663e+00 6.70028 3.07339
9651   2.59373e+00 6.76821 3.64173
9652   2.15930e+00 6.56621 2.90544
9653   7.63521e-01 7.08986 3.12691
9654  -1.38132e+00 7.16422 4.27093
9655   1.78450e+00 6.76471 3.76664
9656   1.26577e+00 6.85132 3.34145
9657   5.80045e+00 6.44959 5.37794
9658   5.07422e+00 6.32934 4.02125
9659   1.35932e+00 6.73688 3.19524
9660   2.59405e+00 6.51257 4.00216
9661   5.52410e+00 6.08240 3.81531
9662   4.77458e+00 6.37889 2.89495
9663  -5.06075e-01 7.34749 3.62313
9664  -6.74481e-01 7.23966 3.38609
9665   5.70586e-01 7.20908 3.77237
9666   1.19806e+00 6.97436 2.85733
9667  -3.21068e-01 7.06671 2.52944
9668   3.52364e+00 6.74997 2.63292
9669   8.88088e-01 6.87439 3.52000
9670   3.87801e+00 6.49266 2.46793
9671   5.26800e+00 6.11323 2.86192
9672   3.68829e+00 6.49890 4.65225
9673   2.67126e+00 6.81106 3.62695
9674  -6.71899e-01 6.63190 7.35045
9675   1.83501e+00 6.88915 3.83799
9676   1.87344e+00 6.73859 2.52433
9677   9.98182e-01 6.85094 3.36851
9678   1.18848e+00 6.86954 3.12003
9679  -3.23147e+00 7.26221 4.63749
9680   7.08480e-01 6.91313 3.44571
9681   1.51997e+00 6.70635 2.96611
9682   1.26760e+00 6.63946 3.02919
9683  -1.27547e+00 6.90125 4.36525
9684   2.86762e+00 6.60265 3.37004
9685   1.55092e+00 6.85832 4.00351
9686   3.82565e+00 6.49089 3.30497
9687  -3.97356e-01 7.08275 3.77168
9688   1.73086e-01 7.10010 3.04501
9689   2.06722e-01 7.08037 3.08254
9690   3.48157e+00 6.61076 3.30887
9691  -2.35709e-02 7.05034 3.29007
9692   1.13441e-01 7.19199 2.47881
9693   1.19464e+00 7.01175 3.54136
9694   2.76045e-01 6.98277 2.97235
9695   3.07248e+00 6.56298 5.06834
9696  -1.19234e+00 7.06397 3.38761
9697  -1.24485e-01 7.14946 3.73608
9698   3.86358e+00 6.57937 3.64315
9699   1.41258e-01 6.93865 3.15286
9700  -2.64486e+00 7.31965 3.12811
9701   2.88513e+00 6.67962 3.52889
9702   2.80804e+00 6.72272 2.35861
9703   1.06278e+00 6.90381 3.06200
9704   2.77877e+00 6.75641 2.51868
9705   4.18319e-01 6.95101 5.04688
9706   5.12451e-01 7.13318 3.00526
9707   2.41585e+00 6.59359 3.81527
9708   2.81386e+00 6.60301 3.05429
9709   2.63769e+00 6.77288 3.54429
9710   1.20806e+00 7.00359 2.47329
9711   3.43622e+00 6.54888 2.96900
9712   3.63023e-01 7.12876 3.52854
9713  -9.29520e-01 7.43007 3.49314
9714   2.03718e+00 6.74473 2.52134
9715   3.78532e-01 6.83367 3.04163
9716   4.57360e-01 6.89961 3.34263
9717   3.77979e+00 6.41773 3.56813
9718   4.48639e+00 6.42751 3.35158
9719   3.96788e-02 7.17887 3.88744
9720   3.11255e+00 6.47237 3.03143
9721   4.41399e+00 6.42458 3.29095
9722   2.54923e+00 6.82455 2.97320
9723   2.10206e+00 6.86837 3.54463
9724  -1.71202e+00 7.10004 4.22049
9725  -1.03834e+00 7.08602 3.05746
9726   1.82253e+00 6.94030 3.31300
9727   1.64985e+00 6.93809 4.14663
9728   1.89486e+00 6.71751 2.70515
9729   7.38595e+00 6.03451 4.40297
9730   2.25187e+00 6.63719 3.06677
9731  -1.35716e+00 7.17004 4.08430
9732   2.34035e+00 6.80133 3.68222
9733   4.31342e-01 6.92132 3.47485
9734   6.51713e-01 7.12758 5.37945
9735   4.66680e-01 6.91841 2.76856
9736   1.45664e+00 6.88762 3.12096
9737   2.40421e+00 6.55810 3.01233
9738   6.99679e-01 7.06031 3.35499
9739  -3.92888e-01 7.06474 3.05615
9740   2.87749e+00 6.71785 3.34936
9741  -1.29803e-01 7.05357 2.79510
9742   4.07323e+00 6.71381 6.80577
9743   3.70322e+00 6.39299 3.15655
9744   5.44598e-01 6.84671 2.52443
9745  -2.98706e+00 7.46338 4.15636
9746   1.62343e+00 6.78830 4.20636
9747   5.88785e+00 6.18490 4.38399
9748   7.03117e-01 7.01360 2.69248
9749   2.74666e-03 6.85421 3.47654
9750   3.75892e-01 7.05444 3.76813
9751   3.09152e+00 6.54590 2.87405
9752   9.61403e-01 6.95251 3.22445
9753   2.05255e+00 6.80088 3.77770
9754   1.42281e+00 6.94488 2.88947
9755   7.34875e-02 6.86962 3.78230
9756  -3.60479e+00 7.45764 3.78813
9757   3.47104e+00 6.63558 3.97368
9758   4.15303e+00 6.45420 2.95639
9759  -4.54579e-01 7.34930 3.59307
9760   2.28443e+00 6.81921 2.47793
9761  -3.54794e+00 7.44439 3.54470
9762   3.76499e+00 6.35970 3.60180
9763   2.05717e+00 6.67729 4.12655
9764   3.82202e+00 6.59818 2.95808
9765   5.82951e-01 6.85468 2.95922
9766   2.67937e+00 6.69272 3.58509
9767   2.16103e-01 6.81407 3.87063
9768   5.01524e+00 6.28925 3.33554
9769   2.84492e-02 6.94707 3.45700
9770   1.44199e+00 6.88893 2.84310
9771   3.79200e-01 6.96736 2.88344
9772   1.65537e+00 6.70850 3.29707
9773  -1.32913e+00 7.23716 3.27398
9774   9.21608e-01 7.10327 2.96596
9775   9.80287e-02 7.24554 3.90684
9776   4.31902e-01 7.06433 3.24249
9777   1.10084e+00 6.91391 3.52608
9778   1.84222e+00 6.84641 4.03571
9779  -1.22155e-01 6.76458 2.98658
9780   2.33462e+00 6.67343 4.44214
9781   2.81502e+00 6.75875 2.90158
9782   1.87575e+00 6.81179 3.96609
9783   3.19114e+00 6.52610 3.90561
9784   2.96435e+00 6.62425 2.97076
9785   3.31221e+00 6.65071 3.30089
9786   4.35291e+00 6.59031 4.21340
9787   5.80954e-01 6.77991 3.41356
9788   1.39295e-01 6.91748 3.88877
9789   2.99336e-01 6.95731 3.73966
9790  -1.15220e+00 7.06275 3.28233
9791   2.87572e+00 6.65373 4.27097
9792   2.81958e+00 6.51068 2.97919
9793   3.52256e+00 6.56957 2.88631
9794   6.63470e-01 6.89878 2.19667
9795   3.58495e+00 6.93704 5.60760
9796   2.09796e+00 6.60760 2.50403
9797   3.59816e+00 6.53560 3.84936
9798   1.48945e+00 6.90827 2.93444
9799   1.22526e+00 6.92259 3.43818
9800   1.60228e+00 6.57339 3.16805
9801   4.30446e-01 7.08954 3.17003
9802   4.35821e+00 6.48692 3.10595
9803   4.64889e+00 6.36241 2.63458
9804   3.63229e+00 6.63024 3.23705
9805   2.04078e+00 6.73671 3.55797
9806  -1.84744e-02 7.11107 3.19102
9807   3.27811e+00 6.60964 2.59554
9808  -8.93493e-01 7.05091 3.64102
9809   1.39582e+00 6.83133 2.83825
9810   3.03791e+00 6.80994 2.73271
9811   8.83197e-01 7.10199 2.78839
9812   1.68984e+00 6.82825 2.91938
9813  -2.16757e+00 7.14281 4.18534
9814   2.69455e+00 6.59151 2.44906
9815   3.28265e+00 6.61660 2.80636
9816   2.49944e+00 6.61916 2.87647
9817   2.73817e+00 6.72004 3.22244
9818   1.22707e+00 7.09078 4.23097
9819  -4.52916e-01 7.06728 4.16472
9820   2.92279e+00 6.68680 4.32509
9821   2.63807e+00 6.67845 2.81819
9822  -1.21490e+00 7.03334 3.13048
9823   7.63643e-01 6.72909 3.34258
9824   2.41333e+00 6.57291 3.24198
9825   1.78596e+00 6.83913 3.44535
9826   4.06435e+00 6.44856 4.26283
9827   3.96249e+00 6.52590 3.27176
9828   1.83107e+00 6.87682 3.16323
9829   1.67366e+00 6.90359 3.31410
9830   4.59674e+00 6.11187 4.07097
9831   3.35827e+00 6.63644 2.76115
9832   2.83557e+00 6.82321 3.17125
9833   2.84023e+00 6.49276 3.99829
9834   1.53125e+00 6.83296 2.71739
9835  -2.53348e-01 7.22594 3.98147
9836   2.29401e+00 6.75850 3.70349
9837   6.98074e-01 6.91516 3.48973
9838   3.65863e+00 6.30506 4.37836
9839   2.61804e+00 6.58130 2.75162
9840   3.67651e+00 6.64412 3.33570
9841   1.19353e+00 6.99955 4.79186
9842   1.23870e+00 6.90928 2.87204
9843   2.25309e+00 6.80852 3.45711
9844   5.46015e-01 6.75676 3.66983
9845   9.85848e+00 5.52129 4.87836
9846  -2.66230e+00 7.58103 4.06835
9847   2.56789e+00 6.94300 3.42817
9848   1.51002e+00 6.62542 3.44724
9849   7.85453e-02 6.98656 3.08432
9850   1.18470e+00 7.00934 3.12513
9851   1.72125e+00 6.83160 4.06580
9852  -1.71117e-01 7.12288 3.47022
9853   2.69223e+00 6.67276 3.38795
9854  -6.32295e-01 7.07366 3.03818
9855   2.29895e+00 6.60417 3.18615
9856   1.98171e+00 6.71100 3.23830
9857  -9.47046e-01 7.35496 3.59538
9858   5.41913e+00 6.07236 3.50052
9859   3.70269e+00 6.40471 3.38538
9860   5.65013e+00 6.25493 2.65846
9861  -1.17004e+00 7.33318 3.19273
9862  -1.04815e+00 7.05046 4.74703
9863   4.27740e+00 6.42631 3.11248
9864   1.66117e+00 6.92385 3.66579
9865   1.31130e+00 6.74059 2.97841
9866  -1.99861e-01 7.04965 3.55220
9867   9.53400e-01 6.98471 2.92903
9868   1.14422e+00 6.90199 3.13226
9869   3.13851e+00 6.78534 3.44061
9870   4.30924e+00 6.23588 2.56061
9871   1.59588e+00 6.92774 2.96060
9872   6.49040e-01 6.77051 3.51154
9873   3.22784e+00 6.60176 2.79492
9874  -2.51733e+00 7.46360 2.76831
9875   2.46683e+00 6.63188 3.38928
9876   1.58644e+00 6.98404 3.32702
9877   5.39119e+00 6.52736 3.94150
9878   2.89288e+00 6.64272 3.12362
9879   4.23977e-01 6.95155 3.29260
9880   4.04090e+00 6.26345 3.73999
9881   3.01288e-02 7.05327 3.95882
9882   1.92216e+00 6.66798 3.31719
9883   1.53774e+00 7.02006 3.77113
9884   3.71753e+00 6.50223 3.00440
9885   1.45794e+00 6.90806 3.21544
9886   3.49483e+00 6.42477 3.73578
9887   7.47599e-01 6.78891 3.65755
9888   2.80488e+00 6.64216 3.81487
9889   3.58182e+00 6.36590 4.76538
9890   1.99123e+00 6.59204 2.81756
9891  -2.86238e-01 6.89248 3.42291
9892   1.27802e+00 6.81369 4.03284
9893   3.20148e+00 6.64360 3.57565
9894   6.18114e+00 6.29564 4.11924
9895   1.80585e+00 6.79471 3.01666
9896   3.87379e+00 6.38308 3.60241
9897   3.12424e+00 6.69855 2.83277
9898   2.28767e+00 6.94257 3.67998
9899   3.53919e+00 6.47655 3.74814
9900   2.39901e+00 6.69479 2.64703
9901   2.76961e+00 6.53943 3.82302
9902   3.02451e+00 6.63942 3.35164
9903  -5.99732e-01 7.11675 3.09070
9904   3.84309e+00 6.44678 3.10244
9905   3.84978e-02 6.97408 2.94663
9906   1.85491e+00 6.60770 3.00332
9907   4.33113e+00 6.59463 3.73099
9908   1.46121e+00 6.82246 2.88823
9909  -1.08304e+00 7.27483 2.85540
9910   4.09944e-02 7.04450 3.53937
9911   6.76281e+00 5.99286 3.11811
9912   2.67064e-01 6.91817 3.51611
9913  -5.44190e-01 7.02995 2.95050
9914   3.60480e+00 6.38170 4.54250
9915   8.09565e-01 6.79926 3.07628
9916   2.89808e+00 6.59946 3.17163
9917   5.81752e+00 6.46764 2.96965
9918  -5.44893e-01 7.12179 3.15077
9919  -1.24005e+00 7.11838 3.36846
9920   2.15281e+00 6.74092 2.70024
9921   9.58581e-01 6.99559 3.95194
9922   3.81951e+00 6.21920 4.89035
9923   2.59788e+00 6.55206 3.43297
9924   3.54683e+00 6.50613 2.98934
9925   6.71867e-01 6.96899 3.16164
9926   1.57215e+00 6.69970 2.65713
9927   6.30286e+00 6.73160 6.11661
9928   1.19873e+00 7.00655 4.51557
9929   6.76076e-01 6.82634 3.49623
9930  -1.04599e+00 7.30547 3.66113
9931   2.32593e+00 6.77207 2.44735
9932   4.23548e-01 7.12268 3.05368
9933   1.34091e+00 6.91550 3.52317
9934   1.27299e+00 6.72911 4.47669
9935   6.26670e-01 6.93452 3.22703
9936   2.43720e-02 7.21408 3.21588
9937   1.80197e+00 6.95206 3.36429
9938  -9.66339e-01 7.06034 3.12296
9939   1.24480e+00 6.90433 4.02123
9940  -2.88629e+00 7.64166 3.24959
9941   1.85926e+00 6.83238 3.41141
9942   9.42621e-01 7.01195 3.00421
9943   1.22618e+00 6.87027 3.79869
9944   6.21598e+00 6.34511 3.03180
9945   1.54169e+00 6.92678 2.91704
9946   4.04967e+00 6.38716 2.68328
9947   7.77060e-01 6.89147 3.11851
9948   3.56907e+00 6.58134 3.54595
9949   2.77592e+00 6.45749 3.29435
9950   3.64428e+00 6.51407 2.99452
9951   1.66116e+00 6.67623 3.31147
9952   1.97475e+00 6.86646 3.03982
9953   4.78492e+00 6.35546 3.45932
9954   2.12059e+00 6.87118 2.73643
9955   5.73504e-01 6.87468 2.38308
9956  -4.56610e+00 7.66580 3.76113
9957  -1.02024e+00 6.93593 4.51179
9958   2.93587e+00 6.50009 3.69278
9959   7.27430e-01 6.92902 3.20793
9960   4.03831e-01 7.02466 3.84322
9961   3.15829e+00 6.68606 3.36467
9962   1.57737e+00 6.81945 3.77637
9963   9.07711e-01 6.95671 3.20398
9964   1.74954e+00 6.77442 3.40600
9965   4.95471e+00 6.14337 3.32339
9966   3.60383e+00 6.65079 4.05002
9967  -1.92462e-02 7.17840 3.12796
9968   2.13757e+00 6.63724 3.63466
9969   1.35241e+00 6.90061 3.18893
9970   1.11961e+00 6.75250 3.66527
9971   3.05806e+00 6.33073 3.72773
9972   3.29238e+00 6.59724 2.60632
9973   5.57805e-01 7.11414 3.34872
9974  -7.27237e-01 7.25671 3.10241
9975   3.31552e+00 6.65625 3.39337
9976  -3.20894e+00 7.37238 3.91828
9977   2.84000e+00 6.65327 3.12514
9978   2.39067e+00 6.70921 3.27227
9979   4.66504e-01 6.98829 3.18132
9980   3.68332e+00 6.52701 2.77147
9981   1.78115e+00 6.69851 4.39853
9982  -2.42963e-01 6.97297 4.58509
9983   2.09331e+00 6.85907 2.60929
9984   3.75971e+00 6.50738 2.64444
9985   4.91186e-01 6.85358 3.88987
9986   2.45853e+00 6.58775 3.17175
9987   3.29100e+00 6.42007 3.21483
9988   2.96516e+00 6.72303 3.41917
9989  -6.43228e-02 7.03402 3.01727
9990   1.83783e+00 6.79711 2.78717
9991   2.18731e+00 6.75823 3.09235
9992   3.31176e+00 6.61065 2.47483
9993   2.89325e+00 6.66041 3.50639
9994   7.76395e-01 7.02009 3.17891
9995   6.66953e+00 6.21860 3.21904
9996   4.07566e+00 6.50769 3.31126
9997   4.75276e+00 6.07765 4.16112
9998   1.31627e+00 6.93987 2.90318
9999   1.54754e+00 6.81638 4.63228
10000  3.03483e+00 6.64369 3.05411

Distributions of posterior parameter estimates

We will learn convenience tools for plotting later.

post |> 
  pivot_longer(cols = everything(),
               names_to = "Parameter",
               values_to = "Estimate") |> 
  ggplot(aes(Estimate)) +
  geom_density() +
  facet_wrap("Parameter", scales = "free")

Distributions of posterior parameter estimates

Modeling categorical predictors

Combinations of 0 and 1
- Multiplication of parameters by 0 drops them
- Induces additional uncertainty
Separate categories
- Preferred
- Estimate means directly
- Code as integers starting with 1 (rethinking, stan) or as factors (brms, rstanarm)

Modeling two category predictors

Generate data:

n = 50
Means = 5 and 8

set.seed(76546)
n <- 50
D <- tibble(y = c(rnorm(n, 5, 1),
                  rnorm(n, 8, 1)),
            Group01 = rep(0:1, each = n),
            Group12 = rep(1:2, each = n))
print(D, n = 100)

Modeling two category predictors

# A tibble: 100 × 3
        y Group01 Group12
    <dbl>   <int>   <int>
  1  5.42       0       1
  2  5.06       0       1
  3  3.69       0       1
  4  4.58       0       1
  5  3.93       0       1
  6  3.55       0       1
  7  5.24       0       1
  8  6.08       0       1
  9  5.78       0       1
 10  5.16       0       1
 11  6.03       0       1
 12  4.49       0       1
 13  5.08       0       1
 14  4.58       0       1
 15  3.59       0       1
 16  5.92       0       1
 17  5.52       0       1
 18  4.25       0       1
 19  3.96       0       1
 20  4.10       0       1
 21  2.46       0       1
 22  4.23       0       1
 23  3.39       0       1
 24  4.09       0       1
 25  6.56       0       1
 26  4.24       0       1
 27  3.61       0       1
 28  5.40       0       1
 29  6.22       0       1
 30  7.21       0       1
 31  6.36       0       1
 32  4.90       0       1
 33  2.81       0       1
 34  4.87       0       1
 35  5.68       0       1
 36  4.17       0       1
 37  4.26       0       1
 38  4.16       0       1
 39  5.77       0       1
 40  4.45       0       1
 41  6.50       0       1
 42  5.52       0       1
 43  5.02       0       1
 44  5.97       0       1
 45  6.27       0       1
 46  3.67       0       1
 47  7.50       0       1
 48  4.53       0       1
 49  3.58       0       1
 50  6.12       0       1
 51  7.31       1       2
 52 10.0        1       2
 53  7.69       1       2
 54  7.46       1       2
 55  6.76       1       2
 56  8.53       1       2
 57  5.84       1       2
 58  7.13       1       2
 59  8.98       1       2
 60  5.63       1       2
 61  7.62       1       2
 62  7.96       1       2
 63  8.07       1       2
 64  5.49       1       2
 65  9.53       1       2
 66  7.79       1       2
 67  6.31       1       2
 68  8.06       1       2
 69  7.45       1       2
 70  8.76       1       2
 71  9.88       1       2
 72  7.65       1       2
 73  8.37       1       2
 74  7.52       1       2
 75  8.67       1       2
 76  7.22       1       2
 77  7.06       1       2
 78 10.6        1       2
 79  7.81       1       2
 80  8.56       1       2
 81  8.93       1       2
 82  7.33       1       2
 83  6.12       1       2
 84  8.34       1       2
 85  5.36       1       2
 86  9.13       1       2
 87  9.24       1       2
 88  7.73       1       2
 89  9.41       1       2
 90  7.08       1       2
 91  7.60       1       2
 92  9.46       1       2
 93  9.39       1       2
 94  6.74       1       2
 95  8.66       1       2
 96  4.32       1       2
 97  7.76       1       2
 98  5.77       1       2
 99  7.85       1       2
100  7.23       1       2

Model statements

$y$ follows a normal distribution with a mean and standard deviation:

\[y \sim Normal(\mu, \sigma)\]

The mean is a linear function of a mean for Group 0 ($Group0$) and an offset for Group 1 ($b1$):

\[\mu = Group0 + b1 \cdot Group01\]

Group01 is either 0 (for group “0”) or 1 (for group “1”)

Prior prediction

Mean for Group 0 probably $0 \pm 10$
Difference for Group 1 probably $0 \pm 6$

set.seed(3479711)

sim_model <- function(ii) {
  Group0 <- rnorm(1, 0, 5)
  Grp1 <- rnorm(1, 0, 3)
  tibble(rep = ii,
         Group01 = rep(0:1, each = n),
         y = Group0 + Grp1 * Group01 + rnorm(n = 50, mean = 0, sd = 1))
}

sims <- purrr::map(.x = 1:500,
            .f = sim_model) |> 
  list_rbind()
print(sims, n = 200)

Prior prediction

# A tibble: 50,000 × 3
      rep Group01       y
    <int>   <int>   <dbl>
  1     1       0 -0.0532
  2     1       0 -2.53  
  3     1       0 -0.800 
  4     1       0 -0.277 
  5     1       0 -0.610 
  6     1       0 -0.292 
  7     1       0 -0.196 
  8     1       0  0.454 
  9     1       0 -0.192 
 10     1       0 -0.715 
 11     1       0 -1.03  
 12     1       0 -1.76  
 13     1       0 -1.36  
 14     1       0 -0.543 
 15     1       0 -0.943 
 16     1       0 -1.28  
 17     1       0 -2.31  
 18     1       0  1.38  
 19     1       0 -0.838 
 20     1       0 -0.857 
 21     1       0 -0.623 
 22     1       0 -0.915 
 23     1       0 -0.159 
 24     1       0 -1.56  
 25     1       0 -1.86  
 26     1       0 -0.759 
 27     1       0 -0.435 
 28     1       0  0.236 
 29     1       0 -1.19  
 30     1       0 -1.59  
 31     1       0 -0.0642
 32     1       0  1.17  
 33     1       0 -0.954 
 34     1       0  0.556 
 35     1       0  1.07  
 36     1       0 -1.28  
 37     1       0  0.706 
 38     1       0 -0.126 
 39     1       0  0.815 
 40     1       0 -0.652 
 41     1       0 -0.385 
 42     1       0 -1.16  
 43     1       0 -0.308 
 44     1       0  0.393 
 45     1       0 -0.806 
 46     1       0  1.26  
 47     1       0  0.809 
 48     1       0  0.195 
 49     1       0  1.65  
 50     1       0 -0.851 
 51     1       1  1.80  
 52     1       1 -0.673 
 53     1       1  1.05  
 54     1       1  1.57  
 55     1       1  1.24  
 56     1       1  1.56  
 57     1       1  1.66  
 58     1       1  2.31  
 59     1       1  1.66  
 60     1       1  1.14  
 61     1       1  0.822 
 62     1       1  0.0886
 63     1       1  0.494 
 64     1       1  1.31  
 65     1       1  0.909 
 66     1       1  0.575 
 67     1       1 -0.454 
 68     1       1  3.23  
 69     1       1  1.01  
 70     1       1  0.995 
 71     1       1  1.23  
 72     1       1  0.936 
 73     1       1  1.69  
 74     1       1  0.287 
 75     1       1 -0.0113
 76     1       1  1.09  
 77     1       1  1.42  
 78     1       1  2.09  
 79     1       1  0.661 
 80     1       1  0.266 
 81     1       1  1.79  
 82     1       1  3.02  
 83     1       1  0.897 
 84     1       1  2.41  
 85     1       1  2.92  
 86     1       1  0.568 
 87     1       1  2.56  
 88     1       1  1.73  
 89     1       1  2.67  
 90     1       1  1.20  
 91     1       1  1.47  
 92     1       1  0.696 
 93     1       1  1.54  
 94     1       1  2.24  
 95     1       1  1.05  
 96     1       1  3.11  
 97     1       1  2.66  
 98     1       1  2.05  
 99     1       1  3.50  
100     1       1  1.00  
101     2       0  1.63  
102     2       0  3.29  
103     2       0  1.60  
104     2       0  1.49  
105     2       0  2.09  
106     2       0  2.26  
107     2       0  2.59  
108     2       0  0.920 
109     2       0  1.93  
110     2       0  3.33  
111     2       0  1.61  
112     2       0  1.59  
113     2       0  2.59  
114     2       0 -0.0354
115     2       0  3.62  
116     2       0  1.75  
117     2       0  2.32  
118     2       0  2.80  
119     2       0  0.708 
120     2       0  2.15  
121     2       0  1.05  
122     2       0  1.72  
123     2       0 -0.562 
124     2       0  0.981 
125     2       0  2.35  
126     2       0 -0.167 
127     2       0  0.295 
128     2       0  0.881 
129     2       0  1.34  
130     2       0 -0.748 
131     2       0  1.07  
132     2       0  1.76  
133     2       0  1.97  
134     2       0  2.48  
135     2       0  0.418 
136     2       0  0.721 
137     2       0  0.426 
138     2       0  1.82  
139     2       0  2.30  
140     2       0  1.10  
141     2       0  1.89  
142     2       0  0.0798
143     2       0  1.08  
144     2       0  2.66  
145     2       0  0.283 
146     2       0  0.526 
147     2       0  1.02  
148     2       0  0.152 
149     2       0  3.35  
150     2       0  2.02  
151     2       1  0.0826
152     2       1  1.74  
153     2       1  0.0524
154     2       1 -0.0641
155     2       1  0.536 
156     2       1  0.707 
157     2       1  1.04  
158     2       1 -0.630 
159     2       1  0.378 
160     2       1  1.78  
161     2       1  0.0648
162     2       1  0.0380
163     2       1  1.04  
164     2       1 -1.59  
165     2       1  2.07  
166     2       1  0.205 
167     2       1  0.766 
168     2       1  1.25  
169     2       1 -0.842 
170     2       1  0.597 
171     2       1 -0.501 
172     2       1  0.171 
173     2       1 -2.11  
174     2       1 -0.569 
175     2       1  0.799 
176     2       1 -1.72  
177     2       1 -1.25  
178     2       1 -0.669 
179     2       1 -0.210 
180     2       1 -2.30  
181     2       1 -0.484 
182     2       1  0.208 
183     2       1  0.422 
184     2       1  0.927 
185     2       1 -1.13  
186     2       1 -0.829 
187     2       1 -1.12  
188     2       1  0.273 
189     2       1  0.751 
190     2       1 -0.447 
191     2       1  0.340 
192     2       1 -1.47  
193     2       1 -0.465 
194     2       1  1.11  
195     2       1 -1.27  
196     2       1 -1.02  
197     2       1 -0.527 
198     2       1 -1.40  
199     2       1  1.80  
200     2       1  0.474 
# … with 49,800 more rows

Plotting the prior predictions

Are the predicted outcomes on the right scale?

sims |> 
  mutate(Group01 = factor(Group01)) |> 
  group_by(rep, Group01) |> 
  summarise(Mean = mean(y)) |> 
  ggplot(aes(x = Group01, y = Mean)) +
  geom_point(position = position_jitter(width = 0.25))

Plotting the prior predictions

Fitting the model

fm <- ulam(
  alist(
    y ~ dnorm(mu, sigma),
    mu <- Group0 + b1 * Group01,
    Group0 ~ dnorm(0, 5),
    b1 ~ dnorm(0, 3),
    sigma ~ dhalfnorm(0, 3)
  ),
  data = D,
  chains = 4,
  iter = 5e3
)

Fitting the model

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Chain 4 finished in 0.2 seconds.

All 4 chains finished successfully.
Mean chain execution time: 0.2 seconds.
Total execution time: 0.9 seconds.

Inspecting the chains: `traceplot`

traceplot(fm)

Inspecting the chains: Rank histogram

trankplot(fm)

Inspecting the output

precis(fm)

           mean         sd     5.5%    94.5%    n_eff    Rhat4
Group0 4.916791 0.17508991 4.636459 5.195120 4577.275 1.000210
b1     2.856348 0.25038747 2.457519 3.258397 4713.830 1.000465
sigma  1.242580 0.09012353 1.107153 1.391474 6156.711 1.000089

Posteriors

Each row is a simultaneous sample of all parameters:

post <- extract.samples(fm) |> 
  as.data.frame()
post

       Group0      b1    sigma
1     5.02212 2.54851 1.194350
2     5.18027 2.59637 1.290330
3     4.87321 2.97078 1.242540
4     5.02465 2.85688 1.274070
5     4.88022 2.70676 1.279630
6     4.88433 2.80594 1.266730
7     5.11331 2.69135 1.199890
8     5.10328 2.55082 1.261210
9     5.14735 2.88616 1.325860
10    4.89626 2.61736 1.306720
11    5.27859 2.49096 1.169050
12    5.06145 2.46621 1.373850
13    4.95710 3.10707 1.155370
14    5.25032 2.78871 1.438360
15    5.01802 3.01507 1.515840
16    4.97745 3.02413 1.398910
17    4.78715 3.34176 1.322520
18    5.05212 2.35885 1.154020
19    4.72610 2.83778 1.118560
20    4.94026 2.96140 1.336650
21    4.80292 2.75294 1.172370
22    4.67587 2.99104 1.183620
23    4.93943 2.80914 1.293050
24    4.94492 3.01715 1.252660
25    4.93815 3.14741 1.515350
26    4.76759 2.92533 1.210850
27    5.38596 2.29561 1.355220
28    4.77912 3.07026 1.221410
29    5.35819 2.72516 1.331800
30    4.50117 3.29396 1.338970
31    4.82969 2.82245 1.490910
32    4.62383 3.19178 1.254440
33    5.06445 2.56408 1.326650
34    4.66597 3.28833 1.318720
35    5.02023 2.66184 1.266340
36    4.71317 3.30100 1.263540
37    5.00101 2.70268 1.264410
38    4.70649 2.70851 1.418800
39    5.07952 2.75177 1.240240
40    5.03487 2.83632 1.221740
41    4.51592 3.64965 1.405170
42    4.95087 3.08947 1.165510
43    5.07716 2.85449 1.255520
44    4.76647 2.76164 1.251580
45    4.95636 2.85019 1.163990
46    4.82046 3.15202 1.199420
47    4.91958 3.06321 1.165650
48    4.84139 2.63306 1.198680
49    5.09280 2.34037 1.239280
50    5.06108 2.96810 1.113750
51    4.88487 2.80556 1.379430
52    5.06860 2.81355 1.160130
53    4.91026 2.85817 1.259460
54    4.90437 2.90722 1.247890
55    4.91762 3.04830 1.272550
56    4.75754 3.03218 1.148170
57    4.59372 3.15644 1.126000
58    4.72396 3.26043 1.334560
59    5.06413 2.86218 1.118270
60    4.84296 3.06229 1.262750
61    4.99426 2.55954 1.318010
62    4.93776 2.66786 1.259710
63    4.85355 2.65683 1.326570
64    5.14635 2.57157 1.144810
65    5.03384 2.95719 1.234600
66    4.81100 3.13836 1.294380
67    4.46560 3.37672 1.246800
68    4.68913 3.37688 1.176490
69    5.05004 2.83340 1.253130
70    4.94768 2.89684 1.300660
71    4.97148 2.93143 1.246250
72    4.40914 3.11162 1.278220
73    4.94613 2.70542 1.258400
74    5.14768 2.78999 1.265610
75    4.98813 2.91612 1.197890
76    4.76274 3.22847 1.264310
77    4.95003 3.08477 1.278500
78    4.89655 3.00162 1.344670
79    5.14723 2.71871 1.120320
80    4.79520 2.78448 1.257710
81    5.02254 2.86022 1.159340
82    4.91826 2.73743 1.219500
83    4.53158 2.92735 1.195020
84    5.25446 2.20146 1.232290
85    4.84266 2.77712 1.193660
86    5.12064 2.68345 1.265490
87    4.67973 3.21202 1.325240
88    5.20377 2.32520 1.206540
89    4.88215 2.89237 1.115820
90    4.86653 3.00180 1.297070
91    4.97499 2.91286 1.270350
92    5.15906 2.90964 1.226830
93    5.02287 2.46419 1.270690
94    5.13488 2.31781 1.310180
95    4.81233 2.91272 1.194930
96    4.93088 2.39457 1.247760
97    4.92623 3.09423 1.198880
98    5.02870 2.63027 1.227240
99    4.96201 2.80823 1.311730
100   4.71177 3.27544 1.609270
101   4.81171 3.12371 1.285940
102   4.85647 3.01900 1.194380
103   4.77235 3.06972 1.216360
104   5.03065 2.50422 1.175240
105   4.81579 2.89715 1.319140
106   4.79050 2.96434 1.232480
107   4.94382 3.18959 1.195320
108   5.07720 2.52761 1.137900
109   4.94297 2.65823 1.187350
110   5.05701 2.96955 1.225760
111   4.66717 3.05881 1.225990
112   4.68400 3.58544 1.313990
113   5.16781 2.66333 1.259500
114   5.06488 2.55583 1.319420
115   5.25933 2.46126 1.208810
116   4.96159 2.72871 1.217430
117   5.07713 2.74102 1.069480
118   4.95812 2.98122 1.153240
119   4.84525 2.84951 1.193640
120   4.89765 2.59489 1.182670
121   4.69400 3.35732 1.286520
122   4.83026 2.76824 1.272210
123   4.83249 2.77781 1.100140
124   4.99173 3.02282 1.244350
125   4.34951 3.71246 1.148240
126   4.80723 2.78687 1.535150
127   4.93416 2.98753 1.230340
128   4.66262 3.32799 1.255490
129   5.40248 2.39028 1.169250
130   5.13008 2.57854 1.171300
131   4.79834 2.82591 1.215890
132   5.07048 2.41570 1.235880
133   5.11364 2.89312 1.257770
134   4.86543 3.14910 1.175210
135   4.55212 2.96657 1.247440
136   4.92250 2.84746 1.172470
137   5.01609 2.64275 1.412060
138   5.07839 2.56454 1.365580
139   4.64023 3.33187 1.368650
140   5.04215 2.92440 1.243270
141   4.73821 3.22105 1.115680
142   5.09386 2.73408 1.289880
143   4.99951 2.99793 1.393310
144   5.34242 2.54669 1.329410
145   4.99299 3.26289 1.315050
146   4.64347 2.91118 1.207470
147   4.72140 3.06354 1.198440
148   4.88484 3.03045 1.156570
149   5.04479 2.77324 1.083670
150   5.33159 2.42817 1.243810
151   4.87418 2.88787 1.420740
152   5.02835 2.77995 1.320740
153   4.81479 3.19452 1.313840
154   4.65747 2.69224 1.166600
155   4.89096 2.65747 1.256800
156   4.89113 2.81217 1.163670
157   4.94420 2.76234 1.319630
158   4.85005 2.98088 1.348640
159   4.88620 3.04033 1.227650
160   4.97018 2.60297 1.232460
161   4.98424 2.87711 1.189900
162   4.81157 3.08180 1.260040
163   5.06419 2.52139 1.155920
164   5.10709 2.66491 1.230830
165   5.21996 2.56646 1.277530
166   4.72507 2.83720 1.218260
167   4.72919 3.18395 1.129300
168   4.72227 2.90112 1.294080
169   4.89885 2.61558 1.240820
170   5.13199 2.68343 1.279870
171   4.91410 2.93026 1.198040
172   4.67409 3.04452 1.425790
173   4.99771 2.76983 1.161490
174   4.64360 3.05858 1.344830
175   4.77374 3.12180 1.239830
176   4.79403 3.03065 1.201910
177   4.90641 3.05972 1.193590
178   4.49470 3.29743 1.254240
179   5.07794 2.95662 1.254900
180   5.00750 2.55717 1.286920
181   4.84271 2.66060 1.265170
182   4.90056 2.77808 1.143240
183   4.80186 3.01573 1.116330
184   4.78758 2.80592 1.261230
185   4.98645 2.96526 1.332620
186   4.80888 2.48274 1.308230
187   5.23432 2.38106 1.333350
188   4.85956 3.24518 1.189170
189   5.34921 2.27853 1.248060
190   5.18151 2.57720 1.211440
191   5.07968 2.78161 1.204730
192   5.04483 2.85293 1.266500
193   4.97577 2.81224 1.065160
194   5.06200 2.53264 1.387560
195   4.98598 2.90586 1.600680
196   4.92355 2.76152 1.203040
197   4.95021 2.75417 1.238420
198   4.90461 2.66142 1.338560
199   4.95585 2.59752 1.174650
200   4.83021 2.80625 1.310810
201   4.67461 2.94742 1.239570
202   4.97475 3.04009 1.236970
203   4.76501 3.01832 1.095330
204   5.00402 2.75176 1.066980
205   5.18344 2.40901 1.311070
206   4.95896 2.87764 1.317350
207   5.17981 2.48443 1.459380
208   4.82097 2.92691 1.285440
209   4.82569 2.95233 1.077730
210   5.00098 2.88532 1.350810
211   5.21501 2.23024 1.340710
212   4.93225 2.77945 1.113340
213   4.64207 3.24606 1.190670
214   4.90159 2.87412 1.265210
215   5.45105 2.34144 1.444610
216   4.99818 2.80331 1.186630
217   5.06436 2.84048 1.199150
218   5.04855 2.66206 1.135210
219   5.04581 2.56869 1.346140
220   5.06566 3.04466 1.296530
221   4.89966 2.67007 1.224100
222   4.88231 2.68556 1.109060
223   4.90492 2.87856 1.180680
224   4.69065 3.25627 1.228500
225   4.79947 2.65809 1.346730
226   4.72231 3.04555 1.226610
227   4.83814 2.65930 1.157570
228   4.68447 3.07095 1.092380
229   5.11470 2.35650 1.390510
230   4.91025 2.86155 1.191510
231   4.94479 2.63973 1.191270
232   4.60149 3.00523 1.272850
233   5.01817 2.79032 1.227740
234   4.93308 2.64674 1.228140
235   4.76884 2.84804 1.256980
236   4.85770 2.43449 1.158080
237   4.92837 2.72396 1.298410
238   5.18782 2.60786 1.110550
239   4.91709 2.92968 1.238060
240   4.89490 2.96465 1.302050
241   4.89554 3.09037 1.216530
242   4.68813 3.22683 1.135600
243   4.73199 3.13412 1.145840
244   4.78159 2.72360 1.274560
245   4.83561 3.01492 1.316080
246   4.98514 2.83840 1.175360
247   4.72027 3.07452 1.118510
248   5.11701 2.59472 1.243680
249   5.01608 2.71152 1.362270
250   4.77822 2.81289 1.331770
251   4.74926 2.64135 1.256150
252   4.83247 2.91145 1.317780
253   4.60189 3.10113 1.248470
254   4.70720 3.14289 1.357530
255   5.07687 2.81619 1.123900
256   4.73708 3.24262 1.231640
257   5.04828 2.89486 1.114700
258   4.50480 3.20884 1.332210
259   4.83563 2.93108 1.220440
260   5.05929 2.47366 1.416800
261   4.97654 2.85276 1.136060
262   4.89316 3.08108 1.214770
263   4.82824 2.96156 1.307580
264   4.41184 3.40007 1.233860
265   4.82552 2.78980 1.249170
266   4.97909 3.01680 1.205790
267   4.76075 3.08204 1.152320
268   4.92406 2.75821 1.196970
269   5.03647 2.75721 1.123650
270   5.19511 2.85422 1.300870
271   4.61110 3.18495 1.268490
272   5.05154 2.56713 1.245510
273   5.08790 2.85987 1.266550
274   4.88604 2.84680 1.306060
275   4.87317 2.99949 1.268890
276   4.99595 2.58519 1.193490
277   4.88098 2.90024 1.107830
278   4.92732 2.87687 1.272170
279   5.12309 2.69761 1.206860
280   5.04634 2.95524 1.242590
281   4.85656 2.64539 1.178400
282   4.72370 3.04806 1.326360
283   4.98263 2.75783 1.264280
284   4.89124 2.74100 1.268710
285   4.84943 2.72081 1.237620
286   4.93866 2.72571 1.261930
287   4.94147 3.17668 1.224940
288   4.93059 2.87852 1.353850
289   4.74917 2.81887 1.213270
290   5.12532 2.80018 1.343530
291   4.83409 3.28515 1.145460
292   4.69191 2.99511 1.277490
293   4.87477 2.71312 1.204210
294   4.80388 2.78258 1.150400
295   5.15950 2.33953 1.421070
296   5.07320 2.84079 1.150010
297   5.16975 2.71531 1.232170
298   4.91908 2.78790 1.164350
299   4.88514 2.89922 0.992143
300   4.65593 3.03184 1.349850
301   4.78351 3.10518 1.409580
302   4.71708 3.04452 1.382780
303   4.77888 3.16145 1.240220
304   5.01279 3.36785 1.239400
305   5.17272 2.59372 1.265230
306   5.26130 2.69031 1.169070
307   5.09079 2.69954 1.415420
308   4.96842 2.68585 1.212420
309   4.98306 2.51042 1.196730
310   4.77487 3.33536 1.284790
311   5.16116 2.68830 1.286780
312   5.23733 2.60982 1.283360
313   4.86228 2.88987 1.226130
314   4.88631 2.84363 1.140950
315   4.81943 2.78210 1.303210
316   5.35268 2.37524 1.342830
317   4.92928 2.93207 1.161930
318   4.98523 2.85828 1.197940
319   4.86437 3.20946 1.214680
320   4.70040 2.92976 1.135290
321   4.84110 3.19447 1.171690
322   4.89260 3.06264 1.315940
323   4.88516 2.75077 1.107270
324   4.65956 3.31147 1.173180
325   4.76597 2.81636 1.390080
326   4.98131 2.82618 1.111290
327   4.94198 3.23531 1.272980
328   4.78130 2.96898 1.118920
329   4.75516 3.03122 1.268980
330   5.18556 2.49565 1.239830
331   4.92905 2.74440 1.345260
332   5.03233 2.57522 1.133410
333   4.98386 2.92011 1.292020
334   4.80613 2.81120 1.182250
335   4.98868 3.20657 1.232910
336   5.02383 2.72440 1.286870
337   4.84685 3.00510 1.370050
338   5.32198 2.47107 1.276210
339   4.78307 3.28421 1.425950
340   5.16140 2.41833 1.204790
341   4.83333 2.70731 1.265730
342   4.70449 3.02095 1.327030
343   4.71165 3.02078 1.353400
344   4.83592 3.29124 1.051690
345   4.77228 3.12497 1.365880
346   4.90838 2.82303 1.355950
347   4.93274 2.62816 1.230860
348   4.85865 2.80014 1.097650
349   4.85896 3.04036 1.185130
350   4.65919 3.17165 1.156050
351   4.82701 2.95195 1.322530
352   4.85196 3.12719 1.272980
353   4.74987 3.17422 1.123220
354   4.93125 2.88325 1.243540
355   4.92065 2.85997 1.079250
356   4.93325 2.83079 1.189590
357   4.97403 2.74516 1.188610
358   4.90759 2.64854 1.136260
359   4.96688 2.48472 1.252910
360   4.99186 3.02063 1.334250
361   4.88976 2.77824 1.175020
362   4.83802 2.98202 1.126770
363   4.88603 2.88827 1.340960
364   4.92462 2.63647 1.382860
365   5.03846 2.63885 1.249450
366   4.61879 3.20293 1.392720
367   4.72697 2.95239 1.220830
368   4.90788 2.69907 1.160970
369   5.31812 2.25849 1.502020
370   4.91357 3.00469 1.241260
371   5.15996 2.42750 1.302520
372   4.99704 3.04838 1.219730
373   4.95945 2.58217 1.278140
374   4.81643 2.85918 1.233330
375   5.00671 2.79432 1.231620
376   4.95055 2.67891 1.276240
377   4.93242 3.04548 1.304620
378   4.81039 3.23404 1.276620
379   4.75083 3.11823 1.277490
380   4.69217 3.23969 1.227760
381   4.97839 2.75084 1.169330
382   4.96224 2.73333 1.299520
383   4.69120 3.00055 1.252370
384   4.95291 2.53350 1.214520
385   5.04228 2.80905 1.258920
386   5.19618 2.28325 1.338910
387   4.69556 3.33641 1.308820
388   4.78229 2.83679 1.140860
389   4.80436 3.04713 1.286180
390   4.90193 3.06616 1.293980
391   4.66054 3.33168 1.248870
392   4.95824 2.70856 1.190120
393   5.21877 2.53185 1.228760
394   5.20630 2.79205 1.214600
395   4.62652 3.13631 1.208180
396   4.97676 2.69058 1.185830
397   5.06108 2.96810 1.113750
398   4.85319 2.89571 1.097280
399   4.98727 2.67467 1.146370
400   4.95022 2.77122 1.158730
401   4.76301 2.95141 1.310950
402   5.11655 2.64265 1.214570
403   4.78849 2.85260 1.124640
404   4.94747 3.01791 1.216140
405   4.94907 2.74314 1.217770
406   5.00610 2.80893 1.250170
407   4.93875 2.94967 1.263170
408   4.72815 3.30533 1.144860
409   4.70261 3.09931 1.199500
410   4.93742 2.88142 1.249950
411   4.79132 3.05458 1.207330
412   4.99015 2.78363 1.152610
413   5.04493 2.83573 1.189110
414   4.90635 2.81087 1.275720
415   5.12992 2.58466 1.168850
416   4.93202 2.89285 1.250210
417   4.60959 2.93160 1.130630
418   5.13046 2.75052 1.174480
419   5.08153 2.91092 1.190840
420   5.06129 2.85348 1.053710
421   5.29684 2.39213 1.269650
422   5.09906 2.49404 1.190450
423   4.86977 2.99370 1.285850
424   4.98947 2.81419 1.096300
425   4.25197 3.69972 1.395510
426   5.14729 3.07542 1.386640
427   4.60964 2.90672 1.226960
428   5.10208 2.68660 1.151170
429   4.84439 2.79454 1.278230
430   4.78720 3.24172 1.286470
431   4.88842 3.49397 1.263530
432   4.94361 2.69468 1.199640
433   4.98039 2.65443 1.334260
434   5.01753 2.89946 1.252510
435   4.92418 2.88402 1.351040
436   4.97502 2.80261 1.283510
437   4.59207 3.22779 1.235320
438   4.66200 2.99212 1.241820
439   4.86342 3.10959 1.379080
440   5.08390 2.81118 1.186860
441   4.82550 3.12699 1.222530
442   4.97574 2.89451 1.258820
443   5.07580 2.65089 1.251390
444   5.23571 2.61397 1.195040
445   4.89413 3.07320 1.119990
446   5.02644 2.66604 1.252850
447   4.74808 3.32276 1.147180
448   4.78354 2.99571 1.211980
449   4.67662 3.18198 1.448260
450   4.71732 3.23051 1.221510
451   4.81888 3.01843 1.332190
452   4.76969 3.08386 1.243660
453   4.65442 3.02852 1.110500
454   4.94020 2.93848 1.093110
455   4.95810 2.75177 1.200600
456   4.83622 3.09645 1.373020
457   4.87956 2.77515 1.217960
458   5.05089 2.50925 1.291370
459   4.45955 3.43978 1.218470
460   5.10357 2.99400 1.141400
461   4.92151 2.58319 1.267850
462   4.92419 2.76863 1.110780
463   5.14728 2.69020 1.281060
464   4.77069 3.17848 1.238380
465   4.88663 2.71955 1.208940
466   5.09953 2.63348 1.080460
467   5.23741 2.63891 1.215480
468   5.03891 2.90499 1.413710
469   4.72898 3.02420 1.212160
470   5.14632 2.77250 1.247970
471   4.64589 3.16361 1.229390
472   4.54960 3.05515 1.124250
473   4.85271 2.64552 1.334830
474   4.93860 2.82214 1.107780
475   4.73992 2.62019 1.042220
476   4.93825 2.91177 1.267040
477   5.31764 2.12503 1.310460
478   4.92639 2.94093 1.260470
479   5.04792 2.63248 1.387570
480   5.03483 2.61914 1.362140
481   5.41156 2.42113 1.319450
482   4.77852 3.01305 1.206690
483   4.53970 3.13444 1.162180
484   4.79800 2.86325 1.132640
485   4.94943 2.71040 1.210330
486   4.91193 2.51600 1.285410
487   4.75973 2.78574 1.181340
488   5.20705 2.80943 1.129300
489   5.27640 2.59498 1.187380
490   5.13555 2.77711 1.131930
491   4.76713 2.94316 1.326880
492   5.05683 2.74430 1.086780
493   4.69479 3.47047 1.271700
494   4.45653 2.94997 1.182810
495   4.79480 3.09650 1.297330
496   4.48664 3.45073 1.200110
497   4.76078 2.92417 1.365730
498   5.15675 2.71133 1.233000
499   4.96847 2.69685 1.207760
500   4.69317 3.05911 1.356870
501   5.00909 2.79563 1.244620
502   5.04223 2.91335 1.362650
503   4.84001 2.62634 1.364910
504   4.95434 3.14839 1.138910
505   4.61479 3.37167 1.239250
506   4.87204 2.88320 1.492300
507   4.87437 2.96132 1.185330
508   5.46064 2.28773 1.428490
509   4.98804 2.87949 1.150460
510   5.07299 2.61240 1.225870
511   5.25155 2.42876 1.491460
512   5.14085 2.60930 1.451820
513   5.32676 2.56725 1.187820
514   5.01627 2.90001 1.265940
515   4.85594 2.85663 1.170260
516   4.91448 2.88232 1.107630
517   5.14591 2.20612 1.339430
518   5.39529 2.58207 1.388300
519   4.92285 2.90814 1.135970
520   4.87132 2.84721 1.268260
521   4.89080 2.68373 1.327570
522   5.03029 2.45665 1.238710
523   4.97684 2.91159 1.113360
524   4.98338 2.77406 1.204540
525   5.06897 2.99616 1.163890
526   4.84813 2.85888 1.267920
527   4.53226 3.21683 1.108130
528   4.80538 3.11134 1.188580
529   4.92588 2.88718 1.160030
530   4.52096 3.41026 1.157370
531   4.91829 2.84926 1.235050
532   5.36533 2.13919 1.105670
533   4.62584 3.31950 1.376810
534   4.80215 2.99401 1.384650
535   4.63623 3.09447 1.096780
536   4.88506 2.88387 1.287370
537   4.88430 2.78999 1.260210
538   4.68778 3.00970 1.141120
539   4.91786 2.61336 1.211440
540   4.74866 2.93278 1.188910
541   4.94937 2.86444 1.201480
542   4.86728 3.00180 1.148100
543   4.85221 2.79139 1.085410
544   5.02145 2.93288 1.129140
545   4.84709 2.80555 1.245900
546   4.98720 2.81428 1.312820
547   4.78766 3.07922 1.150040
548   5.05317 2.98474 1.281390
549   4.81192 3.17182 1.127510
550   4.87548 3.14502 1.288720
551   4.86590 2.84801 1.315720
552   4.92939 2.66255 1.354620
553   4.83023 2.99615 1.137620
554   4.68837 3.10330 1.354010
555   4.74015 2.86738 1.266500
556   4.53332 3.03833 1.242060
557   5.34959 2.29494 1.533080
558   5.13870 2.37687 1.574980
559   4.80330 3.18292 1.268300
560   4.97491 2.65912 1.192380
561   4.83660 3.04720 1.224310
562   5.12515 2.42563 1.242010
563   4.69801 3.10701 1.273560
564   5.08282 2.80937 1.145810
565   4.85396 2.74698 1.224680
566   4.93353 3.31231 1.337560
567   4.99453 2.43573 1.214860
568   4.92033 2.95680 1.175640
569   4.76710 2.77947 1.213120
570   4.78846 2.86199 1.148680
571   5.06229 2.78856 1.203970
572   4.74529 3.03865 1.231560
573   4.65691 3.35952 1.234940
574   4.81913 2.74774 1.227830
575   4.70054 3.07383 1.224620
576   4.97135 2.95265 1.336920
577   4.78805 3.34869 1.152530
578   4.92047 2.88744 1.295290
579   4.83554 2.82331 1.101520
580   4.80320 2.76606 1.395190
581   4.91902 2.99828 1.224330
582   4.95766 3.20102 1.205290
583   5.00262 2.75663 1.037030
584   4.71369 3.04450 1.205290
585   4.79829 2.82887 1.266320
586   5.27027 2.63142 1.216730
587   4.87847 2.87324 1.152000
588   5.04011 2.77474 1.244940
589   4.99069 2.47504 1.137110
590   5.00333 2.71397 1.267480
591   4.62761 3.36769 1.321500
592   4.82662 2.68278 1.269400
593   4.84735 2.94466 1.181090
594   4.75363 3.07973 1.246580
595   4.79889 2.67964 1.149420
596   4.70339 2.91887 1.182770
597   4.86040 3.03381 1.334150
598   4.88123 3.13888 1.155530
599   4.77788 3.25025 1.368310
600   4.84289 2.89556 1.154730
601   4.98078 2.32541 1.349150
602   4.78062 3.00056 1.367920
603   4.81572 2.97583 1.206200
604   4.87627 2.88831 1.231200
605   4.97330 2.91442 1.305600
606   4.70796 3.11930 1.400400
607   4.78241 3.07727 1.173800
608   4.63033 3.33754 1.327780
609   4.95473 3.03131 1.296460
610   4.82824 2.94648 1.148410
611   5.22523 2.17621 1.395020
612   4.75268 2.90702 1.327180
613   4.83824 2.75393 1.157990
614   4.95716 2.80956 1.239420
615   5.08988 2.73109 1.386940
616   4.75864 3.04117 1.291470
617   4.94670 2.99328 1.387710
618   4.70878 3.32662 1.099950
619   5.14807 2.87743 1.246730
620   5.04175 2.88934 1.330560
621   4.77043 3.23871 1.259720
622   4.80265 3.12182 1.362770
623   4.73065 3.30602 1.288720
624   4.93507 2.40655 1.252220
625   5.22722 2.61501 1.161190
626   5.05844 2.38750 1.297070
627   5.22664 2.62839 1.240920
628   4.78309 2.89321 1.127970
629   4.95278 2.71952 1.128140
630   4.85553 2.74962 1.409310
631   4.88277 2.93164 1.257360
632   5.00399 2.86654 1.314160
633   4.55019 3.46179 1.233440
634   4.75434 2.91021 1.345670
635   4.91881 2.97236 1.228620
636   4.87221 2.98731 1.254480
637   4.89286 2.97916 1.380960
638   4.56246 3.26429 1.279520
639   4.67070 3.15386 1.169270
640   5.03996 2.63889 1.143130
641   4.92866 2.85246 1.318980
642   5.06640 2.87631 1.377000
643   4.72712 3.24789 1.283220
644   4.83072 2.85552 1.224610
645   5.25290 2.74606 1.411380
646   4.84587 2.71239 1.202990
647   5.05045 2.41758 1.118290
648   5.09278 2.36800 1.206430
649   4.87895 2.88689 1.417290
650   4.76100 2.86138 1.292670
651   5.11443 2.84906 1.342420
652   4.77951 2.74118 1.546510
653   4.99615 2.43298 1.171050
654   4.84939 3.06826 1.212070
655   5.11054 3.11860 1.329500
656   4.77528 3.11060 1.079710
657   5.03987 2.73125 1.358120
658   4.80141 3.00472 1.440560
659   4.88038 2.88024 1.212040
660   4.80538 3.16715 1.221850
661   5.09149 2.72947 1.265550
662   4.87102 2.62970 1.304600
663   4.95327 2.49283 1.287030
664   4.75183 3.25270 1.188800
665   4.52135 3.06525 1.179700
666   4.61027 3.36153 1.156140
667   4.94345 2.87001 1.187260
668   5.12822 2.85571 1.227860
669   4.95456 2.95796 1.371530
670   4.76465 3.08587 1.278300
671   5.06771 2.70833 1.153980
672   4.88466 2.67133 1.181400
673   4.93090 2.83217 1.167610
674   5.14063 2.54290 1.235220
675   5.23703 2.71794 1.323730
676   4.80277 3.10253 1.232140
677   4.96655 2.89533 1.256240
678   4.89586 3.05698 1.258860
679   5.12186 2.42134 1.295860
680   4.94330 3.24356 1.341900
681   4.78479 2.91004 1.168760
682   4.95331 3.02635 1.051940
683   4.80297 3.45884 1.511560
684   4.99968 2.80229 1.128880
685   5.06432 2.69698 1.115050
686   5.09126 2.73727 1.392370
687   4.75037 2.93796 1.205780
688   5.04995 2.60012 1.398140
689   5.06286 2.80626 1.150380
690   4.70443 3.32526 1.336020
691   4.85999 2.73185 1.121240
692   4.80598 2.97793 1.390040
693   4.89626 2.72651 1.136460
694   4.96692 2.78607 1.109890
695   5.31469 2.57740 1.247430
696   5.00495 2.87205 1.322770
697   5.30824 2.16624 1.166110
698   5.07211 2.58917 1.260770
699   4.96781 2.90794 1.127110
700   5.00002 2.75372 1.228800
701   5.01324 2.95570 1.199260
702   5.00999 2.88293 1.220720
703   5.53247 2.14346 1.297670
704   4.53438 3.20214 1.120830
705   5.09848 2.70197 1.149380
706   5.03522 2.93406 1.230390
707   5.00875 2.56775 1.156610
708   4.97318 2.59800 1.161370
709   4.89291 2.34044 1.304980
710   4.78884 3.07848 1.161510
711   4.97792 3.16253 1.511310
712   4.62298 3.00112 1.369890
713   4.55345 2.86660 1.180950
714   4.82833 3.05852 1.222620
715   4.94958 2.94835 1.230200
716   5.04453 2.70974 1.230350
717   5.18835 2.32216 1.374780
718   4.89317 2.88600 1.189400
719   5.01223 2.54637 1.349130
720   4.65973 3.11549 1.305650
721   4.84939 3.01560 1.206000
722   4.43986 3.43791 1.199750
723   4.77426 3.21200 1.248310
724   5.07536 2.69392 1.118390
725   5.05881 2.62226 1.244500
726   4.85858 2.48158 1.273170
727   5.10274 2.69679 1.158390
728   4.74393 2.82233 1.152860
729   4.72048 2.85334 1.237660
730   4.92538 2.83221 1.087760
731   4.96881 2.89560 1.062320
732   5.23714 2.38272 1.179130
733   4.96593 2.92280 1.227340
734   5.18306 2.60983 1.207060
735   4.87209 2.67618 1.294330
736   4.60508 2.83510 1.324050
737   5.03954 2.54946 1.077820
738   4.85251 3.34963 1.215680
739   5.15109 2.78987 1.226320
740   4.92552 2.68512 1.145950
741   4.97870 2.74972 1.152120
742   4.91506 2.89005 1.245660
743   4.77531 3.04752 1.373590
744   5.02996 2.93009 1.229420
745   4.97898 3.00557 1.246860
746   4.78343 3.07743 1.234190
747   4.61073 3.33158 1.191370
748   4.81308 2.79728 1.221580
749   4.83344 2.99132 1.251460
750   4.93216 2.40643 1.176700
751   4.86728 3.00180 1.148100
752   4.78132 2.85550 1.156820
753   5.17389 2.65412 1.067140
754   4.97579 2.65376 1.107260
755   4.81356 3.12077 1.419670
756   4.60502 3.14146 1.341090
757   4.80696 2.88740 1.321680
758   4.84167 2.84130 1.237680
759   4.78605 3.24360 1.280850
760   4.73752 3.06833 1.207810
761   4.83888 3.13807 1.317420
762   4.72818 3.23794 1.182380
763   4.80003 2.93679 1.346720
764   4.89163 2.61012 1.096530
765   4.90364 2.68647 1.210420
766   4.73584 2.80355 1.260470
767   4.81326 2.68650 1.264810
768   5.05271 2.91714 1.235660
769   5.05311 3.02351 1.154480
770   5.20690 2.43810 1.278120
771   5.14222 2.67982 1.246900
772   4.86889 2.80832 1.137700
773   5.10072 2.76442 1.162220
774   4.92668 2.80216 1.289630
775   4.52704 3.17686 1.306000
776   5.27677 2.72625 1.201800
777   4.81337 2.86157 1.269930
778   4.17420 3.67115 1.385690
779   4.83774 2.59771 1.267190
780   4.94850 2.88410 1.292540
781   4.79216 3.13384 1.195460
782   4.83155 2.94595 1.209630
783   5.05481 2.34309 1.240220
784   4.97119 3.26692 1.326680
785   4.79911 2.90646 1.119370
786   4.85311 2.98415 1.293260
787   5.08993 2.38439 1.187570
788   5.14945 2.48651 1.202070
789   4.66611 3.59241 1.369620
790   5.01092 2.90140 1.243730
791   5.11297 2.90161 1.443540
792   5.31615 2.18212 1.275800
793   4.67033 2.80179 1.374870
794   4.62912 3.38507 1.190910
795   5.03971 2.51364 1.168100
796   5.21344 2.63773 1.273100
797   4.67173 3.19338 1.506710
798   4.97633 3.13838 1.254660
799   4.71906 3.04726 1.332260
800   4.85753 2.87140 1.223370
801   4.85109 3.00240 1.286620
802   5.01906 3.00116 1.510090
803   4.72517 3.15743 1.123330
804   4.97941 2.95658 1.196360
805   4.70664 2.85755 1.104220
806   4.79382 3.09120 1.345750
807   5.06232 2.54333 1.232900
808   5.05381 2.57465 1.289020
809   4.78102 3.11563 1.220300
810   4.98278 2.51289 1.266900
811   5.05460 2.73443 1.239950
812   4.75284 2.97814 1.265800
813   5.16596 2.54090 1.170120
814   5.18425 2.58535 1.222020
815   4.85608 2.96391 1.152720
816   5.03096 2.93803 1.217200
817   4.82483 2.88591 1.124060
818   4.92598 2.80019 1.091640
819   4.60912 3.02653 1.239750
820   4.84563 2.77952 1.207990
821   5.03104 2.97224 1.160050
822   4.88013 2.91665 1.227100
823   4.94543 3.05608 1.230780
824   4.84733 3.13855 1.154190
825   4.96081 2.85318 1.302400
826   4.99557 2.76689 1.126490
827   4.92065 2.76060 1.244730
828   4.72191 3.35143 1.146860
829   4.83502 2.83238 1.115620
830   5.09604 2.82334 1.145950
831   4.69409 3.11536 1.332670
832   5.23326 2.47812 1.237650
833   4.71226 3.01862 1.195040
834   4.58962 3.30781 1.220630
835   4.81443 2.94798 1.242790
836   4.90584 2.87835 1.240060
837   5.19032 2.72590 1.358220
838   4.93510 2.41896 1.273530
839   4.84795 2.84224 1.273750
840   4.85926 3.05460 1.249740
841   4.99359 2.43696 1.351320
842   4.65787 3.08700 1.297690
843   4.89698 2.85954 1.129660
844   5.02818 3.28310 1.354920
845   4.73318 3.12623 1.269500
846   4.87605 3.01577 1.337560
847   5.07455 2.91228 1.256100
848   4.97789 2.90586 1.190990
849   4.58414 3.00767 1.253130
850   4.68747 3.06967 1.203510
851   4.84735 2.98573 1.260370
852   4.91925 3.14352 1.389420
853   4.94735 3.17192 1.211350
854   4.73660 2.96392 1.132290
855   4.57348 2.96665 1.369430
856   4.97326 2.67090 1.173800
857   5.13752 2.50917 1.218220
858   4.94465 2.89845 1.117180
859   4.67645 3.11096 1.188960
860   4.66916 2.98675 1.256480
861   4.88313 3.05317 1.241370
862   4.94626 2.56461 1.300200
863   5.01125 2.65386 1.239430
864   4.82263 2.78536 1.199690
865   5.23271 2.69571 1.174940
866   5.33366 2.34093 1.313920
867   5.19744 2.24617 1.203580
868   5.07581 3.15039 1.418360
869   4.85770 3.12147 1.261520
870   5.17397 2.13307 1.279620
871   4.76074 3.04632 1.208970
872   4.73331 3.13439 1.345440
873   4.94002 3.06394 1.190360
874   4.96126 2.81274 1.153980
875   4.94128 2.84383 1.210550
876   5.14069 2.74861 1.224160
877   5.00082 3.00762 1.227410
878   4.81111 2.81297 1.198920
879   5.06617 2.82875 1.330360
880   4.90082 2.80492 1.162820
881   5.04634 3.00870 1.188040
882   5.38366 2.40740 1.365800
883   5.15912 2.68467 1.226040
884   4.99947 2.82309 1.406820
885   4.67985 3.10886 1.121550
886   5.02357 2.68318 1.160210
887   4.96868 2.61102 1.145370
888   4.78381 2.95473 1.278730
889   5.05559 2.55704 1.300470
890   4.76537 3.01907 1.298460
891   4.77747 2.89592 1.107370
892   4.81971 2.93350 1.345960
893   4.60473 3.31817 1.304610
894   4.87451 2.93457 1.154530
895   5.17387 2.78414 1.336990
896   4.84671 2.82735 1.131320
897   4.76693 3.04792 1.064120
898   4.88981 2.79977 1.332470
899   4.87057 2.78616 1.049590
900   5.01575 2.86791 1.337040
901   5.14784 2.88493 1.237720
902   4.92855 3.01480 1.282860
903   4.89606 3.17842 1.266060
904   4.64288 3.05312 1.101100
905   5.12104 2.64689 1.170460
906   5.14787 2.61539 1.158950
907   4.80780 3.31520 1.345760
908   5.02315 2.59373 1.218980
909   4.90359 3.19457 1.081790
910   4.91004 3.10814 1.274880
911   4.73305 2.87257 1.363790
912   4.96888 2.83894 1.409560
913   4.82588 2.87330 1.268090
914   4.85738 2.80541 1.347720
915   4.88121 2.95516 1.300610
916   4.67710 3.18956 1.261950
917   5.04955 3.00965 1.263300
918   4.89079 2.75574 1.288940
919   5.00340 2.78227 1.362290
920   4.94261 2.93130 1.330660
921   4.90342 2.66012 1.111230
922   4.87460 2.68263 1.202730
923   4.80262 3.09115 1.248020
924   4.88442 2.90888 1.303900
925   5.42894 2.06708 1.347910
926   5.14450 2.55031 1.200600
927   4.97774 2.90008 1.195600
928   4.89708 2.83206 1.219220
929   5.11041 2.61144 1.212070
930   4.85227 2.63260 1.256980
931   4.81840 2.74207 1.207390
932   4.81690 2.94133 1.285600
933   5.08712 2.90480 1.254320
934   5.00973 2.96162 1.164640
935   5.11269 2.60081 1.235470
936   5.12866 2.66883 1.039540
937   5.04699 2.60480 1.216560
938   5.04178 2.74509 1.214410
939   4.67457 3.22935 1.248930
940   4.90914 2.94600 1.289480
941   4.68454 3.22963 1.178230
942   4.90981 2.87783 1.379630
943   4.93256 2.69525 1.304000
944   4.72284 2.99854 1.521460
945   4.95143 2.62964 1.121310
946   4.68181 3.01783 1.199540
947   5.03891 2.44645 1.171000
948   4.86513 2.86624 1.288640
949   4.88634 2.87622 1.242820
950   5.38240 2.32285 1.530460
951   4.92392 3.07823 1.298770
952   5.07882 2.67744 1.407720
953   4.94408 2.98910 1.360420
954   4.93543 2.81298 1.100340
955   4.43595 3.72989 1.340820
956   4.65881 3.07791 1.325310
957   4.70988 3.35921 1.350560
958   4.94501 2.56734 1.161140
959   4.95180 2.71310 1.266000
960   4.50173 3.26514 1.195510
961   5.11158 2.74766 1.176150
962   4.97948 2.93271 1.232280
963   4.60620 3.00516 1.320330
964   4.94412 2.95608 1.282840
965   5.08508 2.88954 1.149340
966   5.25375 2.67819 1.201870
967   4.91687 2.51099 1.262700
968   4.74092 3.21021 1.206850
969   4.82646 2.85761 1.269050
970   4.79103 2.94358 1.165750
971   5.02816 2.63074 1.245120
972   4.66729 3.10156 1.426810
973   4.85631 3.01238 1.126310
974   4.95228 2.97087 1.143010
975   4.90733 2.79734 1.286030
976   4.71829 3.10808 1.174750
977   4.77707 3.03224 1.377490
978   4.93508 3.12660 1.226030
979   4.98985 2.94319 1.034410
980   4.81552 2.90810 1.220150
981   4.79712 3.31851 1.254900
982   4.82107 2.87688 1.354900
983   4.82110 2.89448 1.131300
984   4.86556 2.80656 1.330320
985   4.47094 3.33363 1.211360
986   4.64076 2.95866 1.285000
987   5.22205 2.43006 1.333770
988   5.11927 2.37393 1.369140
989   4.59145 3.24668 1.175010
990   5.04805 2.41290 1.378380
991   5.07278 2.53758 1.201370
992   4.87383 3.22632 1.223040
993   4.77382 3.18804 1.191680
994   5.07494 2.53152 1.286720
995   5.02548 2.94364 1.330730
996   4.80085 3.14441 1.311850
997   4.75995 2.92777 1.198040
998   4.82469 2.57253 1.327650
999   4.96582 2.90248 1.178830
1000  4.67201 3.10173 1.090550
1001  5.21646 2.71749 1.181070
1002  5.25807 2.28113 1.292120
1003  5.27384 2.44993 1.371150
1004  5.18344 2.40901 1.311070
1005  5.22029 2.62936 1.157440
1006  4.79293 2.79536 1.379110
1007  4.88859 2.71840 1.206950
1008  4.50866 3.01781 1.240500
1009  4.97227 2.89104 1.100060
1010  4.83977 2.93286 1.398110
1011  5.01840 3.32138 1.224990
1012  4.74243 3.14620 1.328600
1013  4.50729 2.80643 1.269080
1014  4.97912 2.36189 1.194710
1015  4.84108 2.99707 1.190640
1016  4.96570 2.75833 1.249770
1017  4.78593 2.94186 1.284890
1018  5.07520 2.40333 1.221630
1019  4.63702 3.36016 1.257220
1020  4.87146 3.02536 1.212550
1021  4.73288 3.15134 1.359390
1022  5.16713 2.50885 1.279860
1023  4.87003 2.98545 1.269350
1024  4.67611 3.18220 1.196770
1025  4.76719 2.74891 1.246780
1026  4.71612 3.29993 1.182940
1027  5.12808 2.46636 1.287320
1028  4.79502 2.75762 1.190160
1029  4.92555 2.67947 1.151100
1030  4.88349 3.14632 1.261750
1031  4.95974 2.61569 1.293830
1032  4.85021 2.82302 1.239600
1033  5.20331 2.58444 1.173940
1034  4.81198 3.19214 1.192510
1035  4.97568 2.78537 1.450040
1036  4.86240 2.97858 1.292660
1037  4.73661 3.28401 1.286960
1038  4.47570 3.29096 1.103490
1039  4.84448 3.01051 1.189900
1040  4.99862 2.59019 1.341140
1041  4.96757 2.87868 1.132630
1042  5.10226 2.72491 1.222770
1043  4.75290 3.13767 1.213690
1044  4.81762 2.75149 1.228020
1045  4.89870 2.72974 1.262990
1046  4.99431 2.39892 1.417590
1047  4.71133 3.10593 1.378810
1048  4.81625 2.97021 1.352240
1049  5.12217 2.60275 1.429070
1050  4.87220 2.69993 1.296760
1051  4.96937 2.67504 1.348160
1052  4.65059 3.28198 1.256150
1053  4.88565 2.94876 1.293470
1054  4.70739 2.85581 1.466970
1055  5.05372 2.70900 1.142620
1056  4.90497 2.91890 1.185440
1057  4.76886 2.67920 1.247320
1058  4.94509 2.81729 1.285100
1059  4.81424 2.74459 1.158760
1060  5.08008 3.17832 1.234780
1061  4.90131 2.81631 1.281870
1062  4.59219 2.99454 1.107710
1063  5.14520 2.79508 1.227010
1064  4.94148 2.78309 1.148190
1065  5.05223 2.50824 1.277900
1066  4.87689 2.84370 1.416120
1067  4.74452 3.06554 0.997783
1068  4.76450 2.63054 1.346630
1069  5.14646 2.87341 1.251500
1070  4.71913 3.01496 1.203970
1071  4.85230 2.73909 1.281910
1072  4.91696 2.76786 1.395260
1073  4.89082 2.94476 1.252890
1074  4.50323 3.34953 1.338930
1075  4.85630 2.82495 1.113510
1076  4.81114 2.76133 1.341360
1077  4.97615 3.00958 1.184210
1078  5.05788 2.55542 1.252000
1079  4.83214 2.88403 1.235770
1080  4.46833 3.13713 1.186120
1081  4.55307 3.32209 1.145300
1082  4.81732 2.89048 1.099260
1083  4.87005 2.99791 1.225110
1084  4.80659 2.91444 1.266940
1085  5.00124 2.66815 1.323180
1086  4.95113 2.85356 1.110670
1087  4.95278 2.71952 1.128140
1088  4.71579 2.83718 1.299380
1089  5.40147 2.32326 1.070570
1090  5.04774 2.63557 1.304050
1091  5.05451 2.78461 1.346840
1092  4.85909 2.91511 1.032210
1093  4.79812 3.47999 1.216190
1094  4.68785 3.54388 1.329400
1095  4.65888 3.16634 1.321040
1096  4.79386 2.94809 1.308160
1097  5.04336 2.61156 1.276750
1098  4.98963 2.65425 1.205320
1099  4.97193 3.00362 1.164700
1100  4.64707 3.11520 1.259250
1101  4.90437 2.69293 1.187440
1102  5.19685 2.28473 1.222110
1103  5.02675 2.98690 1.180950
1104  5.11413 2.80716 1.194280
1105  4.87356 2.82142 1.201420
1106  5.32020 2.43150 1.262700
1107  5.26896 2.57709 1.168780
1108  5.03565 2.75719 1.220070
1109  4.89828 2.97313 1.248130
1110  4.92210 2.91201 1.125820
1111  4.89838 2.83974 1.313300
1112  5.07657 2.63487 1.145440
1113  4.98932 2.76098 1.354690
1114  5.13718 2.52049 1.221730
1115  4.96767 2.55184 1.247890
1116  5.02620 2.53234 1.259940
1117  4.90814 2.96381 1.140280
1118  4.97044 2.91406 1.143710
1119  4.96276 2.52494 1.289480
1120  4.75260 3.11601 1.219610
1121  5.29556 2.43844 1.309930
1122  4.74549 3.22209 1.165070
1123  4.70277 3.24974 1.326420
1124  4.90438 2.88090 1.342880
1125  4.81421 3.09906 1.318130
1126  4.71998 2.90333 1.253780
1127  4.97328 2.51523 1.051530
1128  4.90888 3.19818 1.272590
1129  4.75165 2.76334 1.199670
1130  5.06495 2.95715 1.338820
1131  4.95217 2.29434 1.277930
1132  4.82493 2.75101 1.038150
1133  5.05306 2.69230 1.226390
1134  4.78703 2.92600 1.173510
1135  4.87207 2.72922 1.225800
1136  5.02251 3.01595 1.268850
1137  5.07114 3.07269 1.201500
1138  4.95699 2.86168 1.320070
1139  5.17580 2.24299 1.180960
1140  5.05028 2.79193 1.388380
1141  4.93404 3.01249 1.143530
1142  4.67390 3.06513 1.152420
1143  5.08739 2.31865 1.348320
1144  5.06353 2.73248 1.218800
1145  4.79987 3.34142 1.207400
1146  4.92349 2.52568 1.276010
1147  5.21548 3.00603 1.183830
1148  4.97079 2.76098 1.095050
1149  4.93375 2.65894 1.114760
1150  5.06088 2.61049 1.278890
1151  4.86259 2.82963 1.339610
1152  4.72397 3.15398 1.220950
1153  4.78572 2.94228 1.206690
1154  4.74499 2.66125 1.237170
1155  4.88741 2.94898 1.143550
1156  4.92398 2.80868 1.305950
1157  5.16921 2.66725 1.436380
1158  5.16380 2.66678 1.156910
1159  4.99989 2.97154 1.094560
1160  4.80782 2.89810 1.229280
1161  4.68953 2.83389 1.214400
1162  5.20100 2.99207 1.216610
1163  5.17035 2.75684 1.316440
1164  4.94211 2.88238 1.290520
1165  4.69349 3.04903 1.199150
1166  5.16068 2.58413 1.265210
1167  5.18668 2.47940 1.350810
1168  4.91189 3.01866 1.254470
1169  5.25809 2.28266 1.295380
1170  4.78845 2.95166 1.263200
1171  4.91651 3.03299 1.175010
1172  4.88889 2.76609 1.176990
1173  5.11258 2.55055 1.169440
1174  5.14006 2.42181 1.266600
1175  4.79179 2.98680 1.150350
1176  4.91981 2.81444 1.372290
1177  4.89653 2.79829 1.137120
1178  4.94586 2.65645 1.377670
1179  4.87234 3.12427 1.223540
1180  4.72797 3.08919 1.266020
1181  4.90825 2.84710 1.201870
1182  5.08328 2.59056 1.092060
1183  4.90128 2.79185 1.372830
1184  4.62270 3.08948 1.195870
1185  5.01678 2.84300 1.088860
1186  5.28528 2.55236 1.269740
1187  4.82991 2.81846 1.223280
1188  4.93358 2.56903 1.137450
1189  4.75148 2.89829 1.352190
1190  4.86129 2.94588 1.185490
1191  5.05368 2.66486 1.281000
1192  4.84871 2.73401 1.094840
1193  5.29779 2.48367 1.331980
1194  4.84676 3.07341 1.145220
1195  5.15595 2.93252 1.237070
1196  4.70356 3.20600 1.157110
1197  5.06228 2.92215 1.276380
1198  5.00720 3.07708 1.236900
1199  5.02839 2.74339 1.112560
1200  4.71665 3.16969 1.234010
1201  4.77601 3.00685 1.347750
1202  5.01926 2.82182 1.356490
1203  5.20226 2.56303 1.003220
1204  5.20704 2.26470 1.352320
1205  4.99940 2.65843 1.261130
1206  5.15149 2.79582 1.087720
1207  4.85956 2.97802 1.217790
1208  4.31387 3.40192 1.535840
1209  5.09881 2.59010 1.219970
1210  4.68976 2.80619 1.203440
1211  5.14413 2.86689 1.312560
1212  4.77435 3.20778 1.242490
1213  4.99759 3.03878 1.151810
1214  4.65553 3.10645 1.309340
1215  4.99853 2.60900 1.286130
1216  5.03248 2.50617 1.302520
1217  4.96715 2.92552 1.251000
1218  4.79106 2.67018 1.261620
1219  4.91053 2.48975 1.345240
1220  4.87394 2.76795 1.095050
1221  4.81202 2.91318 1.269360
1222  4.86743 2.90983 1.218750
1223  4.73457 3.33168 1.174280
1224  4.78216 2.86230 1.418080
1225  5.13124 2.77375 1.305950
1226  4.90240 2.96683 1.231430
1227  4.89273 2.91782 1.212260
1228  4.70537 3.14032 1.294320
1229  4.64161 3.03507 1.212760
1230  4.71732 3.23051 1.221510
1231  4.95591 2.82977 1.209250
1232  4.52936 3.28797 1.168390
1233  4.76737 2.89951 1.156050
1234  4.83361 2.68615 1.168030
1235  4.98954 2.73754 1.365100
1236  4.99719 2.54265 1.185440
1237  4.83438 3.25282 1.360390
1238  4.68751 2.96690 1.410810
1239  5.04763 2.79841 1.206230
1240  4.79533 2.68252 1.321770
1241  4.74878 2.83871 1.494290
1242  4.74307 3.25872 1.248770
1243  5.19847 2.55788 1.331110
1244  4.62503 2.87322 1.353310
1245  4.99444 2.71071 1.154710
1246  5.11389 2.43470 1.201570
1247  5.12050 2.63488 1.056080
1248  5.08145 2.41507 1.220220
1249  4.84121 2.85614 1.289070
1250  5.03793 2.87499 1.192940
1251  4.89047 2.90946 1.234040
1252  5.03188 2.64919 1.281550
1253  5.20097 2.75810 1.264900
1254  4.74684 3.46809 1.195070
1255  5.19404 2.85724 1.391910
1256  4.84827 3.03799 1.144590
1257  4.96511 2.77529 1.293180
1258  5.01355 3.04328 1.211280
1259  4.85395 2.59934 1.294180
1260  5.06151 2.56711 1.372770
1261  4.78179 3.01320 1.301670
1262  5.13366 2.64785 1.309110
1263  5.08850 2.51882 1.258170
1264  5.00802 2.73389 1.328920
1265  4.70295 2.94907 1.195350
1266  4.86099 3.02027 1.242480
1267  5.32505 2.29272 1.249360
1268  4.72841 3.29417 1.209920
1269  5.02325 3.14445 1.187420
1270  5.05290 2.56710 1.347500
1271  4.90276 2.83804 1.201480
1272  4.72496 3.22219 1.183570
1273  4.83407 2.78743 1.211560
1274  4.92429 3.36888 1.358500
1275  4.63789 3.20553 1.234700
1276  5.06471 2.62989 1.217360
1277  4.93905 2.69365 1.275220
1278  5.01040 2.71660 1.137270
1279  5.00645 2.72372 1.255600
1280  5.14461 2.57160 1.380930
1281  5.20176 2.63141 1.182040
1282  4.68968 3.13582 1.144300
1283  4.75373 3.04905 1.215140
1284  4.99537 2.63757 1.423320
1285  5.18639 2.59491 1.176350
1286  4.86167 2.86591 1.382770
1287  5.03425 2.97959 1.146040
1288  4.84165 2.81706 1.165950
1289  5.04623 2.82502 1.133950
1290  5.07685 2.64585 1.164970
1291  4.63616 3.00265 1.360410
1292  4.80476 2.91086 1.191880
1293  4.97231 2.89864 1.182110
1294  5.02363 3.05853 1.538190
1295  4.76080 2.57783 1.247840
1296  4.97349 3.06379 1.416720
1297  5.19307 2.64630 1.439880
1298  4.87968 3.14698 1.194790
1299  5.04731 3.03814 1.243600
1300  5.00337 2.89167 1.082720
1301  4.82559 3.07924 1.268060
1302  4.96514 2.74491 1.233400
1303  5.25173 2.47435 1.141650
1304  5.19054 2.59068 1.312150
1305  4.85491 2.88344 1.237830
1306  4.89123 3.13916 1.168920
1307  5.05893 3.12982 1.366400
1308  4.66876 3.14887 1.257550
1309  4.91572 2.69203 1.129820
1310  4.96198 2.63956 1.212870
1311  4.66606 3.13486 1.335940
1312  4.87769 3.04682 1.249330
1313  4.93564 2.56175 1.336680
1314  4.65371 3.12319 1.346490
1315  4.71274 2.93602 1.141100
1316  4.98465 2.60428 1.315120
1317  4.87410 2.95629 1.253990
1318  5.02082 2.82975 1.358860
1319  4.69882 3.01573 1.180040
1320  4.74963 3.08818 1.357640
1321  4.93402 2.83688 1.316860
1322  4.71273 2.95501 1.195180
1323  5.12052 2.81200 1.338550
1324  4.86684 2.78053 1.266250
1325  4.80336 3.06558 1.222460
1326  4.89394 2.87096 1.277610
1327  4.77738 2.95709 1.208050
1328  4.94801 2.83130 1.400020
1329  4.82796 2.77505 1.164410
1330  5.12472 2.53163 1.217950
1331  4.84134 2.81237 1.194280
1332  4.99740 2.85453 1.307580
1333  5.03788 2.49001 1.177040
1334  4.80246 2.67523 1.140130
1335  4.97638 3.03594 1.149890
1336  5.15173 2.55390 1.173940
1337  4.96453 2.56954 1.229750
1338  5.02022 2.63005 1.218930
1339  4.96229 2.67417 1.274210
1340  4.77789 3.06217 1.321900
1341  4.35390 3.22432 1.115870
1342  4.89599 2.85874 1.097720
1343  4.87071 2.88534 1.172470
1344  4.97280 2.73448 1.351340
1345  4.79782 2.94701 1.126400
1346  4.88207 3.01399 1.199990
1347  4.94820 3.36379 1.352810
1348  4.93362 2.72799 1.274470
1349  4.68142 3.23135 1.024990
1350  5.08290 2.48239 1.313530
1351  4.67769 3.01360 1.264900
1352  4.95782 2.83384 1.162060
1353  4.81750 3.12662 1.242580
1354  4.46304 3.14522 1.170600
1355  4.91601 3.23975 1.249170
1356  4.92963 2.89378 1.167440
1357  5.07188 2.51393 1.221880
1358  4.85502 2.74096 1.458810
1359  4.79284 3.12959 1.380270
1360  4.82603 2.79587 1.421810
1361  4.93062 2.97860 1.454620
1362  5.07334 2.58847 1.295950
1363  5.17329 2.64586 1.353830
1364  4.98546 2.76224 1.109510
1365  5.05205 2.47975 1.158060
1366  4.84207 2.98028 1.343170
1367  5.18631 2.60550 1.322050
1368  4.98485 3.00364 1.288900
1369  4.74354 3.26131 1.406760
1370  4.88223 3.07933 1.270930
1371  5.09211 2.47901 1.320950
1372  5.06494 2.70412 1.176590
1373  4.77758 2.90368 1.311800
1374  4.63507 3.27331 1.305280
1375  5.04038 2.47295 1.334100
1376  4.80397 2.86417 1.156290
1377  4.82552 3.03889 1.167260
1378  4.93979 2.91619 1.152380
1379  4.65099 3.08523 1.285740
1380  4.96360 2.49733 1.344420
1381  4.90009 2.70914 1.117720
1382  4.94264 2.95311 1.375310
1383  4.86945 2.88334 1.175610
1384  4.73455 3.05113 1.258450
1385  4.93305 2.73412 1.212780
1386  4.75773 2.88459 1.207450
1387  4.95505 3.38237 1.277720
1388  4.79378 2.76814 1.124320
1389  5.09033 3.36220 1.360610
1390  5.08290 2.55621 1.390560
1391  4.70905 2.85928 1.086280
1392  5.19880 2.51071 1.247980
1393  5.03284 2.80477 1.072910
1394  4.81183 3.04270 1.218850
1395  5.12531 2.32300 1.222960
1396  4.82666 3.06397 1.246780
1397  4.52252 3.03194 1.382350
1398  5.05401 2.55626 1.120670
1399  4.99222 2.94702 1.261960
1400  4.86213 2.82188 1.352140
1401  4.63271 2.94179 1.366290
1402  4.94849 2.88866 1.219360
1403  4.97559 2.58913 1.309300
1404  5.06310 2.84439 1.200740
1405  4.87820 3.38575 1.155800
1406  5.01685 2.95554 1.431460
1407  4.85354 2.72317 1.075140
1408  4.53815 3.40780 1.201560
1409  4.65662 2.94016 1.246040
1410  4.78035 2.98175 1.273920
1411  5.01931 2.51408 1.155650
1412  4.75987 3.09884 1.461590
1413  5.04406 2.68762 1.087880
1414  5.32537 2.17573 1.357200
1415  4.91792 2.60248 1.340490
1416  5.01375 2.87672 1.069790
1417  4.76588 2.61546 1.317780
1418  4.72976 2.74852 1.237600
1419  5.09455 2.54914 1.193310
1420  5.01250 2.72289 1.206900
1421  4.73645 2.76130 1.347200
1422  4.99691 2.89651 1.234120
1423  5.14935 2.47533 1.243000
1424  4.98970 2.77033 1.187730
1425  4.90710 2.65547 1.259770
1426  5.05750 2.80628 1.229900
1427  4.93585 2.96399 1.357490
1428  4.85995 3.19580 1.311710
1429  4.77871 3.29646 1.140170
1430  4.69474 2.99175 1.282640
1431  4.40914 3.11162 1.278220
1432  5.02429 2.32503 1.103200
1433  5.08700 2.53985 1.305440
1434  4.81097 2.85830 1.227870
1435  4.92082 2.88462 1.201530
1436  4.79923 2.85913 1.237030
1437  5.19853 2.63948 1.474940
1438  4.75701 2.97769 1.253260
1439  5.06577 2.31440 1.277980
1440  4.78750 2.94512 1.181550
1441  4.87097 3.09050 1.263720
1442  5.02127 3.10073 1.298060
1443  4.70131 3.35211 1.444400
1444  4.46729 3.25946 1.197080
1445  4.77074 2.73362 1.304610
1446  5.10493 2.85561 1.172190
1447  4.86011 2.99148 1.217250
1448  5.00032 2.93182 1.337000
1449  4.93278 2.45148 1.412750
1450  4.65156 3.11862 1.418080
1451  5.13514 2.63613 1.239490
1452  4.89181 2.87751 1.311180
1453  4.72971 3.05528 1.278340
1454  5.06245 2.58254 1.130250
1455  5.09721 2.65783 1.185810
1456  4.97107 2.78534 1.184760
1457  4.81539 3.08218 1.254030
1458  4.76797 3.16445 1.168070
1459  4.86271 2.87341 1.284600
1460  4.66119 3.01438 1.164900
1461  5.11531 2.56100 1.198690
1462  4.62979 2.90318 1.567880
1463  5.15295 2.93750 1.118470
1464  4.66322 2.81182 1.180280
1465  4.90931 2.88321 1.187960
1466  4.67934 2.99197 1.216800
1467  5.15482 2.66253 1.313480
1468  5.09573 2.78654 1.154970
1469  5.09953 2.61337 1.233210
1470  4.69427 3.11847 1.252890
1471  5.03147 2.70550 1.333660
1472  5.04797 3.05939 1.485740
1473  4.99188 2.51349 1.234210
1474  4.89819 2.89799 1.390350
1475  5.00469 2.87211 1.324730
1476  4.81244 2.93932 1.136070
1477  4.79522 3.04698 1.149870
1478  4.69261 2.93380 1.382370
1479  4.99835 2.52564 1.221850
1480  4.68204 3.13495 1.194710
1481  4.69438 3.21609 1.270130
1482  4.84481 3.07635 1.341030
1483  4.83659 2.84266 1.152790
1484  4.66293 3.01388 1.426230
1485  4.84296 3.06229 1.262750
1486  4.90613 2.69765 1.232120
1487  4.47452 3.43250 1.263230
1488  5.04301 2.75780 1.226790
1489  4.67186 3.03666 1.126710
1490  4.82246 3.31200 1.381950
1491  5.07032 2.62486 1.147270
1492  5.02647 2.71104 1.148180
1493  5.43784 2.23876 1.131960
1494  5.11966 2.69534 1.154290
1495  4.75877 2.98917 1.336770
1496  4.99843 2.75751 1.335990
1497  4.96445 3.04440 1.220730
1498  4.77373 3.16428 1.238010
1499  4.93876 2.88668 1.213890
1500  4.90358 2.55374 1.245610
1501  5.18058 2.47458 1.319750
1502  4.38631 3.22656 1.378960
1503  5.04640 3.15328 1.388910
1504  4.91925 2.89576 1.243260
1505  4.91805 2.93870 1.154320
1506  4.99334 2.93560 1.204320
1507  4.88480 2.64505 1.323030
1508  5.06874 2.66609 1.198150
1509  4.70582 3.07259 1.234660
1510  4.93330 2.66733 1.493520
1511  5.07536 2.61556 1.173320
1512  4.83169 2.89915 1.357950
1513  4.89033 2.87017 1.216000
1514  4.81502 2.96667 1.267920
1515  5.11055 2.64444 1.205530
1516  4.60621 3.39485 1.337560
1517  4.97307 2.72069 1.249970
1518  5.03107 2.89322 1.186050
1519  5.01692 2.89292 1.174780
1520  5.22193 2.55013 1.147890
1521  4.91830 2.71255 1.235780
1522  4.75097 2.86669 1.168820
1523  4.85137 3.04312 1.215810
1524  5.06374 2.84125 1.407140
1525  4.92853 2.90739 1.334060
1526  5.04619 2.57640 1.391070
1527  4.88764 2.81243 1.173620
1528  5.11207 2.59180 1.350610
1529  4.68914 3.11295 1.274090
1530  4.92387 2.77779 1.148360
1531  4.81962 2.89130 1.352960
1532  4.94596 2.75354 1.381570
1533  4.78840 3.35064 1.319000
1534  4.86953 2.87687 1.260100
1535  4.86966 2.64144 1.211410
1536  4.51959 3.85814 1.509680
1537  4.92350 2.92716 1.302450
1538  4.85012 3.04455 1.218240
1539  5.09809 2.45777 1.245740
1540  4.59489 3.22150 1.269600
1541  5.08427 2.58167 1.121180
1542  4.98331 2.65014 1.211050
1543  4.82807 3.12682 1.193430
1544  4.83391 3.12975 1.227170
1545  4.79953 3.04385 1.294780
1546  5.10479 2.62915 1.221290
1547  4.91362 2.81476 1.189420
1548  5.28376 2.65702 1.292040
1549  5.01022 2.47377 1.306660
1550  5.01014 2.80363 1.276230
1551  5.02107 2.66475 1.192130
1552  4.80865 2.97396 1.224430
1553  4.76216 3.06434 1.294780
1554  4.95692 2.48960 1.148500
1555  4.82359 3.07593 1.148570
1556  4.72200 2.79297 1.413400
1557  4.73106 3.00103 1.080560
1558  5.11501 2.71255 1.162090
1559  4.88608 3.16657 1.249600
1560  5.20822 2.94303 1.281840
1561  4.84251 2.87521 1.020090
1562  4.99680 2.61140 1.104720
1563  5.10131 2.71458 1.404650
1564  4.97991 2.60322 1.215660
1565  4.96910 2.71288 1.314370
1566  4.85235 2.88616 1.329280
1567  4.53987 3.18372 1.223370
1568  4.92095 2.74323 1.166580
1569  5.21219 2.57900 1.300930
1570  4.84055 3.05924 1.278430
1571  5.11419 2.78423 1.113810
1572  4.98088 3.09976 1.388650
1573  4.94764 2.85619 1.418980
1574  4.92889 2.78647 1.371220
1575  4.99051 2.90733 1.281370
1576  5.02638 2.45595 1.236770
1577  4.55257 3.13650 1.167610
1578  4.79100 2.90659 1.233780
1579  4.62525 2.89092 1.314700
1580  4.90624 2.98415 1.202030
1581  5.02828 2.57809 1.271270
1582  4.71621 2.89186 1.265830
1583  4.73807 3.01638 1.270050
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1587  5.16312 2.91124 1.238670
1588  4.81103 3.01053 1.255240
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1592  5.05892 2.86082 1.152480
1593  4.63545 3.28445 1.365170
1594  4.87271 2.74553 1.289250
1595  4.89293 2.81598 1.212060
1596  4.92972 2.83942 1.112620
1597  4.57644 3.23874 1.189780
1598  4.62970 3.18617 1.285990
1599  4.80722 2.61098 1.245030
1600  4.85489 2.91789 1.245840
1601  4.91320 2.83196 1.197840
1602  4.67557 3.24843 1.160820
1603  4.73957 2.85618 1.181560
1604  5.10328 2.55082 1.261210
1605  5.27281 2.80974 1.262930
1606  5.04350 2.83656 1.303330
1607  4.80709 2.98330 1.253150
1608  4.98960 2.58534 1.184930
1609  4.89298 3.21742 1.118490
1610  4.70458 3.41271 1.118990
1611  5.08897 3.02195 1.381660
1612  4.96756 2.47943 1.114970
1613  4.76990 2.77717 1.149170
1614  4.96762 2.69469 1.201160
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1616  4.91685 2.54783 1.398440
1617  4.96728 2.78578 1.117850
1618  4.70528 3.27906 1.076340
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1620  4.71656 2.87007 1.158520
1621  4.74404 2.91038 1.397430
1622  4.50370 3.27638 1.298300
1623  4.63111 3.02255 1.174410
1624  5.06005 2.85166 1.228050
1625  4.84996 2.93442 1.365740
1626  4.90394 3.10469 1.051700
1627  4.91302 2.86066 1.218160
1628  4.97462 2.65579 1.403640
1629  4.92135 3.03531 1.052290
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1631  4.73334 3.10096 1.281190
1632  4.86479 2.79543 1.239210
1633  4.81908 2.89587 1.263980
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1636  5.09345 2.77651 1.259520
1637  4.89630 2.72898 1.304160
1638  5.11420 2.92783 1.204840
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1641  4.93291 3.20703 1.381760
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1643  4.80684 3.12103 1.249980
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1700  4.96497 2.87044 1.181540
1701  5.07066 2.84386 1.106840
1702  4.69899 3.29465 1.160460
1703  4.60786 2.99340 1.167720
1704  4.77445 3.01457 1.236100
1705  4.75220 3.15958 1.332900
1706  5.02246 2.63378 1.173330
1707  5.03723 2.72318 1.235770
1708  4.72207 3.23049 1.329700
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1729  4.96122 2.62550 1.215320
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1773  5.25205 2.29104 1.331940
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1775  4.83755 2.93284 1.294050
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1780  4.88679 2.83506 1.418200
1781  4.90946 3.05570 1.224720
1782  5.12111 2.63185 1.445170
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1784  5.13568 2.69862 1.305060
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1793  4.98304 2.76483 1.305430
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1795  4.89981 3.03676 1.209630
1796  4.93432 2.94761 1.356080
1797  4.99866 2.69296 1.277050
1798  4.60959 2.93160 1.130630
1799  4.72585 2.82122 1.340530
1800  4.32016 3.38809 1.498850
1801  4.73941 2.93489 1.444420
1802  4.90293 3.05300 1.118590
1803  5.32944 2.67196 1.398260
1804  4.69265 2.93396 1.148650
1805  4.98912 2.88323 1.340280
1806  4.74502 3.04958 1.511550
1807  4.81308 2.72904 1.284200
1808  4.87858 3.00151 1.267620
1809  4.79083 3.13192 1.216980
1810  4.91648 2.75524 1.128990
1811  5.00996 2.69782 1.163920
1812  4.67657 3.13449 1.158120
1813  4.82436 3.25162 1.111460
1814  4.92111 3.00845 1.139620
1815  5.07340 3.13943 1.455580
1816  4.97373 2.61592 1.303070
1817  5.19180 2.49106 1.168730
1818  5.23717 2.55160 1.164380
1819  5.22523 2.63726 1.259120
1820  4.79669 3.28711 1.210640
1821  4.64020 2.99070 1.351320
1822  4.50661 3.52721 1.261860
1823  5.08552 2.88660 1.300210
1824  4.87907 2.76649 1.049960
1825  5.13731 2.91138 1.242140
1826  5.07531 2.51704 1.430980
1827  4.95079 2.69852 1.206970
1828  5.04429 2.64117 1.176650
1829  4.87421 2.74630 1.214380
1830  4.68749 3.13214 1.241150
1831  5.03855 2.63714 1.152160
1832  4.63784 2.96960 1.305760
1833  4.95079 2.89297 1.343350
1834  4.66510 3.05601 1.229970
1835  4.72421 2.58665 1.254140
1836  4.78463 3.16105 1.041000
1837  4.79183 2.89803 1.206660
1838  4.80675 3.14855 1.211470
1839  4.94242 3.09294 1.364000
1840  5.16034 2.50301 1.345050
1841  5.21742 2.52208 1.285240
1842  4.72186 3.07638 1.207110
1843  5.14438 2.88794 1.235680
1844  4.56156 3.42476 1.261040
1845  4.92729 2.74714 1.077630
1846  4.99243 2.90534 1.123320
1847  4.80385 2.95039 1.195110
1848  5.14902 2.97590 1.289930
1849  4.95348 2.53594 1.304200
1850  4.55421 3.08383 1.236970
1851  4.95596 2.62381 1.191680
1852  4.78345 2.80917 1.441750
1853  4.75327 2.97207 1.213810
1854  4.78814 2.95326 1.023640
1855  4.87011 2.70795 1.272840
1856  4.99423 2.83585 1.235030
1857  4.85625 2.83009 1.311700
1858  5.11369 2.44537 1.221030
1859  4.86883 2.44086 1.501100
1860  5.00795 2.53054 1.202220
1861  5.17544 2.69757 1.149660
1862  5.15969 2.53267 1.060820
1863  4.97071 2.49433 1.439150
1864  4.55810 3.14105 1.303150
1865  4.62980 3.16873 1.226040
1866  5.18257 2.65866 1.219060
1867  4.94502 2.94946 1.164140
1868  4.92584 2.75382 1.227870
1869  4.92422 3.09228 1.379320
1870  4.92386 2.87685 1.179580
1871  4.86522 2.99358 1.209250
1872  4.70791 3.22382 1.347540
1873  4.87155 2.99694 1.443150
1874  5.14815 2.58729 1.296600
1875  4.76277 3.20721 1.246560
1876  4.81888 2.73003 1.373580
1877  4.99230 2.91548 1.236470
1878  4.87337 3.00669 1.317410
1879  4.84761 2.98337 1.358430
1880  5.23184 2.55072 1.215020
1881  4.73626 2.77631 1.264110
1882  4.99701 2.50688 1.282970
1883  4.98735 2.79857 1.312530
1884  5.01842 2.82358 1.112450
1885  4.67573 3.16574 1.192240
1886  4.88822 3.06872 1.182300
1887  4.78244 2.96292 1.290240
1888  4.89853 3.00043 1.181280
1889  4.92117 2.69531 1.131790
1890  4.77019 3.06485 1.198900
1891  5.18385 2.84073 1.316310
1892  5.03009 2.67453 1.440040
1893  4.94249 2.67336 1.214190
1894  4.99298 2.81181 1.167610
1895  4.81319 2.94547 1.375000
1896  5.20338 2.34000 1.137360
1897  5.23014 2.47909 1.429120
1898  4.99717 3.10668 1.186060
1899  4.92020 2.75715 1.313220
1900  4.75974 2.96931 1.259680
1901  4.86103 3.04498 1.183710
1902  4.74886 2.99824 1.230500
1903  4.99413 3.01278 1.323610
1904  4.54814 3.54958 1.194150
1905  4.77884 2.97472 1.240230
1906  4.73908 2.67037 1.300940
1907  4.84541 2.96093 1.028470
1908  5.10228 2.90549 1.355640
1909  4.92381 2.86924 1.223840
1910  5.22451 2.51640 1.153870
1911  5.06535 2.68539 1.109390
1912  4.95885 2.81348 1.219540
1913  5.10589 2.71959 1.311550
1914  4.88510 2.74132 1.165790
1915  4.99739 2.57494 1.104970
1916  5.38183 2.41812 1.256250
1917  5.02149 2.57167 1.294960
1918  4.92764 2.92723 1.211150
1919  4.69020 3.06080 1.324520
1920  4.87581 2.88050 1.361640
1921  5.35491 2.35178 1.430120
1922  4.65107 2.99928 1.166330
1923  4.94772 3.14122 1.215440
1924  4.75598 2.94663 1.208810
1925  4.95003 2.94663 1.233230
1926  5.06338 2.54570 1.496460
1927  4.64477 3.04714 1.187110
1928  4.81451 2.58543 1.145700
1929  4.86610 2.70879 1.278300
1930  5.02123 2.98885 1.326020
1931  4.80689 3.01677 1.283910
1932  5.05928 2.85503 1.218680
1933  4.73876 2.94741 1.114800
1934  4.93499 3.00897 1.157840
1935  4.90955 2.71902 1.185860
1936  4.79363 2.38960 1.249130
1937  4.67833 3.23302 1.168410
1938  4.93272 2.97346 1.243250
1939  4.85147 2.99735 1.187420
1940  4.84750 2.98443 1.297070
1941  4.89656 2.90503 1.073390
1942  5.08171 2.30217 1.171720
1943  4.81723 2.79651 1.240630
1944  4.95484 3.01606 1.252400
1945  4.78798 3.27806 1.290280
1946  5.48980 1.97580 1.464790
1947  5.06494 2.70412 1.176590
1948  4.99025 2.86065 1.271100
1949  4.69435 2.97823 1.404740
1950  5.09999 2.93042 1.259060
1951  5.11408 2.26040 1.421650
1952  4.80328 2.89873 1.137010
1953  4.78112 3.15523 1.079510
1954  4.85915 2.84510 1.224440
1955  4.92613 3.15442 1.513900
1956  4.92313 2.78732 1.243900
1957  4.87699 3.08477 1.184040
1958  4.80664 2.99459 1.173380
1959  5.13679 2.16259 1.269370
1960  4.89538 3.00036 1.216320
1961  4.98214 2.42244 1.140640
1962  5.12283 2.67752 1.138040
1963  4.82734 3.35696 1.232320
1964  4.69644 3.08468 1.173820
1965  5.07433 2.43332 1.331020
1966  4.95262 2.85685 1.205710
1967  5.04428 2.92474 1.156520
1968  5.10384 2.81055 1.088580
1969  4.99420 2.77067 1.210980
1970  4.83831 2.85115 1.163020
1971  5.00249 3.02280 1.203570
1972  5.00778 2.91593 1.024910
1973  4.91502 2.97534 1.174590
1974  5.14643 2.80304 1.196640
1975  4.68940 3.03501 1.268940
1976  4.61208 3.49978 1.413860
1977  5.28028 2.27491 1.278320
1978  5.01554 2.63735 1.240860
1979  4.95778 3.03599 1.362310
1980  5.09305 2.63479 1.403880
1981  5.23907 2.57275 1.240340
1982  4.98382 2.66567 1.258230
1983  4.87573 2.76713 1.271850
1984  4.76154 2.72367 1.221790
1985  5.12442 2.50636 1.134410
1986  4.85235 2.95740 1.354990
1987  5.02860 2.75708 1.181480
1988  5.23561 2.39095 1.094110
1989  4.78128 2.81818 1.175120
1990  5.17740 2.77563 1.221590
1991  4.90528 3.07897 1.207560
1992  4.94215 3.04181 1.193470
1993  5.30045 2.49872 1.267370
1994  5.00010 2.78339 1.210160
1995  4.91760 2.98319 1.303770
1996  4.90050 3.04857 1.286730
1997  5.02468 2.75088 1.169980
1998  5.03045 2.56568 1.351150
1999  4.89069 3.16243 1.225490
2000  4.77143 2.77572 1.189740
2001  5.19579 2.76549 1.293910
2002  5.00967 2.84656 1.432900
2003  4.78407 2.74548 1.218800
2004  5.08460 2.48502 1.098250
2005  5.08814 2.86237 1.123610
2006  4.90989 2.59742 1.182780
2007  5.49302 2.31829 1.444800
2008  5.05477 2.89837 1.140420
2009  4.84203 2.91029 1.369850
2010  4.84532 2.73062 1.174430
2011  4.85534 2.86757 1.294730
2012  4.70473 3.17429 1.203330
2013  4.96793 2.68236 1.215500
2014  4.99519 2.47480 1.280900
2015  4.89519 2.86645 1.255220
2016  4.83914 3.04295 1.307180
2017  4.90190 2.99447 1.357670
2018  4.97726 2.77330 1.287010
2019  5.29906 2.44741 1.251230
2020  4.73926 3.01035 1.192790
2021  5.18692 2.95776 1.406310
2022  4.76795 2.85338 1.123120
2023  5.08002 2.65128 1.277240
2024  5.04300 2.88870 1.264630
2025  4.78838 2.73189 1.225200
2026  4.67084 3.28026 1.330640
2027  4.87456 2.89879 1.257320
2028  4.95848 3.01709 1.257280
2029  4.88504 2.78496 1.092700
2030  5.02525 2.67701 1.240260
2031  4.66937 3.00086 1.191990
2032  4.96275 2.73370 1.488180
2033  5.22928 2.69821 1.155410
2034  4.94369 2.96319 1.299960
2035  5.00285 2.93013 1.191950
2036  4.86312 2.86544 1.091530
2037  4.86804 2.57021 1.533420
2038  5.08679 2.90103 1.272430
2039  5.03870 2.59275 1.237870
2040  5.26712 2.84902 1.329890
2041  4.90015 2.80618 1.441030
2042  5.03166 2.46115 1.173150
2043  5.07330 2.48072 1.198690
2044  4.88699 3.22607 1.228780
2045  5.20800 2.52210 1.343110
2046  4.86468 2.95851 1.221930
2047  4.65218 3.38381 1.225520
2048  4.92723 3.09918 1.384980
2049  5.07179 2.53240 1.128160
2050  4.89645 2.90587 1.275930
2051  4.70806 3.11686 1.181500
2052  4.81244 2.70967 1.166620
2053  4.94824 3.18380 1.261850
2054  4.82780 2.86484 1.155110
2055  4.89314 3.18459 1.264220
2056  5.16104 2.77446 1.216350
2057  4.79132 2.87805 1.165640
2058  5.03746 2.33079 1.338840
2059  4.78870 2.75911 1.158450
2060  5.09293 2.83626 1.281520
2061  4.77836 3.07498 1.272460
2062  4.94927 2.83170 1.155760
2063  5.12878 2.60474 1.184950
2064  4.81395 3.15905 1.172620
2065  4.75352 3.04798 1.232740
2066  5.09898 2.65007 1.168000
2067  5.04272 2.80757 1.121970
2068  5.16374 2.77677 1.140250
2069  4.66931 3.00722 1.315930
2070  5.08754 2.87824 1.248860
2071  4.60853 2.97441 1.405850
2072  4.80309 3.04029 1.103090
2073  4.97582 2.69316 1.198110
2074  4.90437 2.90722 1.247890
2075  4.84316 2.75471 1.121030
2076  5.17826 2.68720 1.150410
2077  4.89718 2.90763 1.081010
2078  4.85999 3.01950 1.223560
2079  4.94092 3.02494 1.319920
2080  4.64038 3.07402 1.235380
2081  5.22024 2.51828 1.436220
2082  4.91167 2.92754 1.231730
2083  4.68823 3.44326 1.170260
2084  4.78607 3.16983 1.083530
2085  4.92678 2.79992 1.152100
2086  5.09277 2.58833 1.169320
2087  4.87101 2.86107 1.238780
2088  4.95898 2.98985 1.138220
2089  4.77492 2.95003 1.159860
2090  5.10394 2.90761 1.370620
2091  4.67672 3.00565 1.416530
2092  5.36744 2.28265 1.084080
2093  5.03561 2.99855 1.270950
2094  4.93424 2.74520 1.156400
2095  5.14934 2.64374 1.223510
2096  4.92182 2.85550 1.271170
2097  5.17807 2.34639 1.231600
2098  5.23388 2.74294 1.403720
2099  4.93801 2.63261 1.189550
2100  4.94527 2.89994 1.254460
2101  4.81022 3.08529 1.143670
2102  5.06857 2.74268 1.247050
2103  4.85360 2.99025 1.140760
2104  4.84753 3.22049 1.265200
2105  5.09435 2.61004 1.221970
2106  4.76193 3.06837 1.189770
2107  4.84574 3.21118 1.175480
2108  4.82880 2.77210 1.287880
2109  4.78025 3.23703 1.289830
2110  4.69898 3.03091 1.231780
2111  5.00351 2.68919 1.160000
2112  4.82402 3.21156 1.220200
2113  4.77248 3.24351 1.198570
2114  4.59879 2.85557 1.462170
2115  4.60689 3.39900 1.127100
2116  4.95989 2.68397 1.250640
2117  4.87857 2.74014 1.190630
2118  5.05282 3.06802 1.193340
2119  5.07887 2.60737 1.299790
2120  4.82002 2.85090 1.143880
2121  4.89032 2.70791 1.191980
2122  4.37646 3.50963 1.252770
2123  4.98756 2.95505 1.395350
2124  4.63646 3.42220 1.178050
2125  5.03714 2.93456 1.363620
2126  4.94678 2.51768 1.232910
2127  4.94879 3.07761 1.138680
2128  4.64749 3.07937 1.311990
2129  4.79025 2.74661 1.300160
2130  4.86636 2.86416 1.096250
2131  4.93674 2.87248 1.324530
2132  4.91309 2.87372 1.299530
2133  4.51709 3.24998 1.183120
2134  4.83905 2.69132 1.253770
2135  4.82095 2.75782 1.170780
2136  5.01501 2.92530 1.334070
2137  4.62897 2.91517 1.205040
2138  5.22368 2.85037 1.203650
2139  5.00075 2.61227 1.242120
2140  4.87186 2.90985 1.219870
2141  4.97229 2.86280 1.278280
2142  4.93457 2.79175 1.351570
2143  4.85529 2.69057 1.348110
2144  4.96754 3.12825 1.224660
2145  4.91347 2.86430 1.133530
2146  4.78930 2.82749 1.247330
2147  4.97074 2.82869 1.276760
2148  4.94706 2.98664 1.235690
2149  5.19388 2.69370 1.062380
2150  4.68099 3.14891 1.355110
2151  5.02626 2.84370 1.106200
2152  5.09656 2.30912 1.283580
2153  4.64116 2.89367 1.335460
2154  5.06073 2.53466 1.168810
2155  4.97645 2.74481 1.162270
2156  4.66344 3.10507 1.089940
2157  4.76723 3.33263 1.303280
2158  5.03675 2.81649 1.205110
2159  5.06583 2.76330 1.235390
2160  4.86333 2.82615 1.079470
2161  5.03119 2.80607 1.204370
2162  5.14986 2.70119 1.152710
2163  4.80623 2.88278 1.297160
2164  4.84083 3.07567 1.008180
2165  4.91185 3.01034 1.279490
2166  4.89349 3.24629 1.375770
2167  5.07706 2.67280 1.160280
2168  5.25059 2.47944 1.309440
2169  5.07820 2.70597 1.214100
2170  5.06069 2.69674 1.293950
2171  4.84848 2.79924 1.220160
2172  4.63850 3.46596 1.258900
2173  4.91237 2.92965 1.280790
2174  5.08714 2.63750 1.357250
2175  5.10946 2.63196 1.276970
2176  4.64550 3.18386 1.228740
2177  4.98899 2.70510 1.172220
2178  4.78812 2.97514 1.274930
2179  4.93721 2.97796 1.131550
2180  4.98705 2.84865 1.298350
2181  4.97283 2.66918 1.090750
2182  5.04010 2.67012 1.116470
2183  4.67356 2.97370 1.165980
2184  5.02242 2.75191 1.433200
2185  4.96518 2.65660 1.308700
2186  4.75667 3.08001 1.214730
2187  4.69365 3.25586 1.167850
2188  5.01114 2.89751 1.175820
2189  4.86606 2.98038 1.221930
2190  5.19352 2.85007 1.262700
2191  4.67566 2.98710 1.237440
2192  4.76839 2.76466 1.251180
2193  5.19961 2.54295 1.457030
2194  4.79904 3.00433 1.275270
2195  5.00778 2.69998 1.132600
2196  4.73314 2.83819 1.097920
2197  5.13790 2.68316 1.215800
2198  4.89885 2.95398 1.214740
2199  4.81236 2.76811 1.157050
2200  4.73469 3.08892 1.254950
2201  4.93410 2.68572 1.220460
2202  5.04817 2.89261 1.248530
2203  4.97007 3.14088 1.286290
2204  5.00589 2.51586 1.271300
2205  4.93484 2.99743 1.335910
2206  5.02429 2.81299 1.253160
2207  4.69541 3.14123 1.191680
2208  5.02719 2.71594 1.234270
2209  4.83143 2.64976 1.215270
2210  4.95859 2.73637 1.083910
2211  4.80405 2.72068 1.282860
2212  4.98416 2.96508 1.182030
2213  4.76836 3.26877 1.311710
2214  4.77361 3.08403 1.293900
2215  4.82752 2.91543 1.239950
2216  4.70947 3.44639 1.228870
2217  4.88763 2.77207 1.226090
2218  4.92282 2.84515 1.149180
2219  4.73599 2.91516 1.334270
2220  4.97112 2.73970 1.361710
2221  4.84919 3.00967 1.241430
2222  4.73645 3.29773 1.333860
2223  4.69741 3.35989 1.407260
2224  4.95427 2.72818 1.148330
2225  4.75886 2.62540 1.274120
2226  4.65863 3.30152 1.208960
2227  4.90392 2.91755 1.199710
2228  5.07008 2.96186 1.174250
2229  5.02791 3.00242 1.177810
2230  4.93436 2.84069 1.161730
2231  5.24378 2.48088 1.154120
2232  5.02467 2.32666 1.346280
2233  4.66026 2.96233 1.125150
2234  4.72082 2.93073 1.319410
2235  4.63046 3.25647 1.104060
2236  4.94451 2.60394 1.220560
2237  4.85757 3.18305 1.316890
2238  5.03575 2.51990 1.280080
2239  4.86070 2.95261 1.414740
2240  4.93802 2.62632 1.349400
2241  4.87653 2.90959 1.213950
2242  4.67897 3.03529 1.238260
2243  5.07209 2.36663 1.121390
2244  4.75294 2.92257 1.361280
2245  4.78431 2.84950 1.173000
2246  4.77231 2.65312 1.215920
2247  5.08003 2.80718 1.186100
2248  5.14992 2.50732 1.179440
2249  5.29066 2.58149 1.234510
2250  4.93135 2.81481 1.294130
2251  5.00236 2.74237 1.157700
2252  5.02505 2.68453 1.110340
2253  4.92988 2.85781 1.247830
2254  5.19511 2.79007 1.162280
2255  5.01077 2.82549 1.299880
2256  4.84905 3.05015 1.242250
2257  4.82027 2.90942 1.132740
2258  5.02444 2.75175 1.238320
2259  4.98098 2.58948 1.391310
2260  5.01542 2.86809 1.201820
2261  4.85878 3.06477 1.088110
2262  4.77206 3.40485 1.278990
2263  5.07010 2.67489 1.081820
2264  4.85036 2.95326 1.200320
2265  4.71351 2.99076 1.245350
2266  5.05509 3.14788 1.234330
2267  4.72626 3.18010 1.157170
2268  4.75056 2.96554 1.266800
2269  4.85764 3.05733 1.266690
2270  4.91887 3.09422 1.320710
2271  4.92683 2.49646 1.359780
2272  4.56765 3.14984 1.329360
2273  4.94024 2.70387 1.256880
2274  4.88856 3.06393 1.184220
2275  5.14942 2.82383 1.245750
2276  5.01761 2.63928 1.229260
2277  5.08132 2.73603 1.260630
2278  4.87875 2.74977 1.256550
2279  4.80139 3.07348 1.553270
2280  4.75341 2.70429 1.179730
2281  4.90947 2.84149 1.219860
2282  4.90366 2.76408 1.226890
2283  5.07783 2.83725 1.345820
2284  4.63717 3.03467 1.181760
2285  4.73273 3.08902 1.241110
2286  5.07681 2.55700 1.203270
2287  4.95240 2.42040 1.262980
2288  4.88402 2.91622 1.244400
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2740  4.92115 2.87353 1.191400
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3602  4.92181 2.91218 1.358330
3603  4.87209 2.86426 1.429700
3604  5.08317 2.97403 1.270010
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3607  5.19124 2.22594 1.133490
3608  5.07757 2.71570 1.072250
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3614  4.66719 3.10102 1.341850
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3623  4.87966 2.82600 1.214770
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3627  4.83781 2.93692 1.087240
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3631  5.09595 2.97028 1.188380
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3633  4.56630 3.10304 1.246570
3634  5.12634 2.58643 1.232590
3635  4.75760 3.16477 1.263070
3636  4.71368 3.23817 1.159180
3637  4.79174 2.92855 1.100840
3638  4.81235 3.10680 1.190120
3639  5.22818 2.78456 1.269530
3640  4.95945 2.92410 1.061520
3641  4.69581 3.24999 1.279220
3642  5.20421 2.57526 1.115200
3643  4.71486 3.18078 1.204450
3644  4.98254 3.14060 1.249120
3645  4.90153 2.65028 1.193400
3646  5.00040 2.79257 1.266210
3647  5.04228 2.64133 1.350860
3648  4.91868 3.01639 1.248260
3649  5.04328 2.33135 1.209520
3650  5.20636 2.61161 1.272760
3651  5.12623 2.44452 1.122020
3652  4.95202 2.70022 1.130520
3653  4.88399 2.95457 1.114130
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3655  4.68404 3.25122 1.204200
3656  4.90737 3.05933 1.176260
3657  4.82022 2.84523 1.167590
3658  5.05086 2.71142 1.146070
3659  5.10635 2.60296 1.088660
3660  4.96812 2.91767 1.302830
3661  4.87474 3.09175 1.140340
3662  4.95266 2.77163 1.277890
3663  4.96153 2.95403 1.196380
3664  5.12851 2.85904 1.369940
3665  5.05449 2.78254 1.193950
3666  5.05513 2.88608 1.221910
3667  5.04337 2.68317 1.205860
3668  5.01579 2.58711 1.098140
3669  4.80392 2.76466 1.354540
3670  4.80095 2.80928 1.043510
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3677  4.70508 2.82129 1.264630
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3680  4.70289 3.11204 1.133350
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3700  4.71753 3.09613 1.184750
3701  4.99097 3.09459 1.261340
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3703  4.92833 2.80508 1.222510
3704  4.85068 2.89834 1.277600
3705  5.06541 2.47836 1.278670
3706  5.24584 2.25342 1.346270
3707  4.93991 2.77531 1.150930
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3709  4.77748 3.20705 1.364780
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3711  5.02295 2.40552 1.289120
3712  4.90020 3.00463 1.385090
3713  5.04291 2.55691 1.241450
3714  4.99216 2.82758 1.238080
3715  4.93750 3.22015 1.158590
3716  4.75274 2.82934 1.376990
3717  4.61310 3.21201 1.356500
3718  4.93035 2.72056 1.251760
3719  4.90593 2.97136 1.338160
3720  5.06461 2.56005 1.251610
3721  4.52862 3.52395 1.339790
3722  5.10639 2.59854 1.271070
3723  5.20222 2.56792 1.173960
3724  5.15500 2.69817 1.215530
3725  4.99818 3.04759 1.208230
3726  4.97275 2.99289 1.510850
3727  5.23354 2.20796 1.293810
3728  4.92886 2.54229 1.151930
3729  4.95332 2.71876 1.204130
3730  5.17545 2.93923 1.367920
3731  4.98197 2.63686 1.157460
3732  5.05239 2.57183 1.022690
3733  5.09844 2.44075 1.186120
3734  4.78683 3.11310 1.261420
3735  5.03755 2.93041 1.325460
3736  5.26829 2.55501 1.384230
3737  4.84195 2.81632 1.283280
3738  5.03508 2.97663 1.185630
3739  4.61862 3.33924 1.060010
3740  4.83399 3.28127 1.234040
3741  4.79723 3.07064 1.246970
3742  4.77634 2.69451 1.273600
3743  4.86358 2.87523 1.202920
3744  4.95560 3.04882 1.155110
3745  4.88340 2.85391 1.194500
3746  4.77119 3.07157 1.170940
3747  4.73829 2.97270 1.259130
3748  5.07033 2.73962 1.141730
3749  4.78997 3.13167 1.091430
3750  5.00160 2.75796 1.247320
3751  4.80150 3.23867 1.248500
3752  4.71858 3.05784 1.224470
3753  4.93198 2.95114 1.259820
3754  4.98402 2.98087 1.186480
3755  4.88304 2.78109 1.131330
3756  4.73844 3.06439 1.297010
3757  4.86159 2.90640 1.355070
3758  5.00266 2.85224 1.319650
3759  4.83363 2.97056 1.326800
3760  4.64805 3.13935 1.262480
3761  4.90501 2.73569 1.392980
3762  5.07274 2.88404 1.095230
3763  4.95447 2.82406 1.247680
3764  5.15918 2.86788 1.176030
3765  4.89402 2.92827 1.216230
3766  4.72581 3.04990 1.376240
3767  4.70467 3.15259 1.260900
3768  4.84853 2.82395 1.281680
3769  4.70396 2.81029 1.319250
3770  4.85413 3.16820 1.153710
3771  4.72488 3.33307 1.294280
3772  4.83735 2.48341 1.259740
3773  4.77334 2.98999 1.110790
3774  5.11834 2.70708 1.374010
3775  5.25551 2.87210 1.374000
3776  5.00419 3.04516 1.172090
3777  4.84871 2.88929 1.251760
3778  5.20797 2.67105 1.257800
3779  4.97772 2.63652 1.239410
3780  5.03091 2.68795 1.129350
3781  4.90896 2.82671 1.262590
3782  4.80500 3.04219 1.224090
3783  4.91723 2.97391 1.110850
3784  5.02558 2.97435 1.387810
3785  4.69470 3.02456 1.317770
3786  4.89701 2.70285 1.210810
3787  5.04423 2.68369 1.129160
3788  4.74353 2.77672 1.073870
3789  5.25103 2.65996 1.240410
3790  4.71347 2.83264 1.273490
3791  4.88798 3.11406 1.257290
3792  4.68159 3.39132 1.165200
3793  5.13203 2.74673 1.281390
3794  4.96399 2.73027 1.415300
3795  4.73778 2.85476 1.382860
3796  4.96436 2.69246 1.349500
3797  5.01496 2.81423 1.225670
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3799  4.70140 3.21641 1.145360
3800  4.98085 2.50377 1.180380
3801  4.89045 2.94878 1.225010
3802  4.85374 3.21288 1.273920
3803  5.09996 2.64554 1.244730
3804  5.05987 2.74580 1.293620
3805  4.70369 3.31471 1.268480
3806  4.77803 2.90855 1.222030
3807  4.95153 2.76790 1.244220
3808  4.85197 3.00951 1.253240
3809  4.88711 3.00638 1.192670
3810  5.08060 2.57388 1.269230
3811  5.01267 2.79720 1.314400
3812  4.58877 3.32343 1.225720
3813  4.77592 2.93714 1.275870
3814  5.03818 2.38791 1.233070
3815  4.37575 3.58342 1.340640
3816  4.75413 2.94017 1.239260
3817  5.02901 2.66016 1.145340
3818  4.93787 2.75015 1.097650
3819  4.58022 3.49834 1.406820
3820  4.82334 3.11902 1.140180
3821  5.10481 2.72402 1.167730
3822  4.94258 2.72105 1.229010
3823  5.25946 2.38832 1.172260
3824  4.94220 2.90101 1.216300
3825  5.21684 2.48667 1.254920
3826  4.99604 2.64833 1.225100
3827  4.86488 2.94182 1.124870
3828  4.73467 2.94031 1.279470
3829  4.93240 2.81737 1.184160
3830  5.09105 2.68523 1.332610
3831  5.06899 2.70549 1.153860
3832  5.09302 2.56571 1.218900
3833  4.79010 2.84751 1.203010
3834  5.06549 2.76003 1.211720
3835  4.75863 3.04234 1.342190
3836  4.87172 3.21558 1.274220
3837  5.01172 3.34053 1.268790
3838  5.13780 2.59601 1.154980
3839  5.27618 2.64812 1.258050
3840  4.91425 2.68570 1.238150
3841  4.86449 2.86679 1.210720
3842  5.18643 2.35946 1.313270
3843  4.87767 2.70286 1.130920
3844  5.10176 2.25216 1.294260
3845  4.98146 2.69435 1.243680
3846  5.03559 2.72639 1.115660
3847  4.80477 2.69330 1.291740
3848  5.12293 2.33993 1.306220
3849  4.64765 3.13214 1.099070
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3852  4.88194 2.87633 1.295500
3853  4.69187 2.94851 1.302110
3854  5.08070 2.69769 1.416080
3855  5.03423 2.54446 1.331820
3856  4.92841 2.99177 1.117190
3857  5.24506 2.78172 1.325600
3858  4.79508 2.84067 1.266280
3859  4.97417 2.92981 1.368810
3860  4.76428 3.07693 1.177200
3861  4.85142 2.88721 1.259630
3862  5.20414 2.44606 1.056110
3863  4.96622 2.80368 1.267460
3864  4.85770 2.65613 1.275990
3865  4.64125 3.43836 1.193500
3866  4.80503 3.13218 1.175340
3867  4.98339 2.49882 1.439450
3868  4.62296 3.28380 1.361340
3869  4.95135 3.02512 1.394310
3870  4.86742 2.66176 1.301790
3871  4.76399 3.19679 1.227960
3872  5.27811 2.49807 1.380530
3873  4.97580 2.88068 1.213070
3874  4.91305 2.34962 1.185280
3875  4.91833 3.06389 1.153960
3876  4.68722 2.80026 1.181570
3877  4.94229 2.75646 1.137110
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3882  5.07754 2.60657 1.228250
3883  4.93258 3.33880 1.195440
3884  5.08151 2.47922 1.083120
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3886  4.66189 2.81179 1.188780
3887  5.15030 2.43538 1.220300
3888  4.85468 2.91019 1.324650
3889  4.88470 3.00699 1.218210
3890  5.21369 2.36195 1.210920
3891  4.93309 2.86785 1.102800
3892  4.90199 3.15853 1.313150
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3894  4.90894 3.10898 1.384960
3895  5.12159 2.59762 1.199320
3896  5.03495 2.89995 1.153050
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3898  4.97621 2.94651 1.252770
3899  4.57151 3.08944 1.209600
3900  4.86830 2.95332 1.456940
3901  4.93183 3.15749 1.211250
3902  4.79730 2.95910 1.215180
3903  4.86205 2.87578 1.151530
3904  4.88787 3.04843 1.268270
3905  4.88686 2.79211 1.198460
3906  5.25060 2.63418 1.328700
3907  4.81394 3.00685 1.220450
3908  5.12962 2.54742 1.137420
3909  4.89615 3.17956 1.166610
3910  4.91353 2.54825 1.215110
3911  4.60813 3.44535 1.331860
3912  4.98021 2.54012 1.273270
3913  4.98449 2.99884 1.354120
3914  5.00136 2.76439 1.268980
3915  5.17756 2.34340 1.239610
3916  4.97701 2.90679 1.186420
3917  4.85050 2.63346 1.379150
3918  5.04371 3.23038 1.129160
3919  4.97035 2.80568 1.089600
3920  5.09239 2.98583 1.334940
3921  4.89225 3.02526 1.204650
3922  5.05135 2.76094 1.256270
3923  4.77813 2.90434 1.149250
3924  4.85271 2.93312 1.201220
3925  4.93796 2.98203 1.238600
3926  5.08145 2.66346 1.259340
3927  5.07759 2.62649 1.134710
3928  4.91889 2.81678 1.238450
3929  5.02370 2.67737 0.998992
3930  4.60994 3.23663 1.153630
3931  4.64507 3.06529 1.310650
3932  5.01410 2.83279 1.228890
3933  4.81235 2.91353 1.307260
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3935  5.21234 2.39731 1.384130
3936  4.78495 3.07253 1.156490
3937  4.76179 2.89249 1.260980
3938  5.18778 2.64472 1.324320
3939  5.35578 2.41127 1.275720
3940  5.29013 2.54747 1.149960
3941  4.95276 3.00796 1.383540
3942  4.99828 2.83861 1.268480
3943  5.03599 2.58827 1.235900
3944  5.06016 2.44217 1.297500
3945  4.96435 2.78051 1.230890
3946  4.77924 3.31324 1.182000
3947  4.70083 2.89614 1.308660
3948  5.05431 2.72148 1.218730
3949  4.84852 3.08517 1.123790
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3951  5.27800 2.51415 1.264430
3952  4.90902 3.00882 1.166540
3953  4.82839 3.17660 1.207840
3954  4.97300 2.68450 1.263750
3955  5.00323 3.18446 1.475890
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3957  4.99771 2.72771 1.307210
3958  4.90100 2.82724 1.256670
3959  4.64289 3.15817 1.075220
3960  4.94126 3.03078 1.067490
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3963  4.98935 2.90696 1.215510
3964  5.01421 2.58835 1.153590
3965  4.81538 2.93410 1.190270
3966  4.93162 2.74492 1.254270
3967  5.00543 2.88020 1.143380
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3991  4.96260 2.71039 1.253640
3992  5.17080 2.54709 1.138840
3993  4.72976 3.22933 1.370090
3994  5.17610 2.20383 1.172010
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4008  4.78465 2.60981 1.280410
4009  5.04324 2.76500 1.206810
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4023  5.04629 2.65054 1.161640
4024  5.13738 2.50152 1.252950
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4026  4.96432 2.44745 1.370620
4027  4.80536 2.99468 1.099030
4028  5.00747 2.77762 1.306550
4029  4.87410 3.07428 1.273910
4030  4.79645 3.22901 1.347970
4031  5.00064 2.74745 1.183730
4032  5.06149 2.74540 1.251510
4033  4.90156 2.86477 1.210810
4034  4.86431 2.93714 1.187460
4035  4.81435 3.00411 1.240680
4036  4.66343 3.27528 1.191100
4037  4.78934 3.25884 1.188480
4038  4.88888 2.83750 1.093330
4039  4.81834 3.11437 1.233170
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4041  5.08090 2.80637 1.190840
4042  4.99285 2.89167 1.229590
4043  4.95470 2.69387 1.174970
4044  5.15245 2.51105 1.335540
4045  4.92620 2.94949 1.277020
4046  4.78763 2.97662 1.320440
4047  4.66300 2.96599 1.185380
4048  4.86312 2.88621 1.105040
4049  4.83875 3.00618 1.281710
4050  5.08490 2.38166 1.446200
4051  5.12969 2.85439 1.227370
4052  5.03045 2.73756 1.193960
4053  5.16939 2.34709 1.272490
4054  4.88180 2.91180 1.380770
4055  5.08107 2.71277 1.272770
4056  4.82883 2.99928 1.225660
4057  4.81972 2.95318 1.173630
4058  4.93503 2.81523 1.209330
4059  4.66960 2.94665 1.148590
4060  5.08996 2.82482 1.217420
4061  4.84563 2.78129 1.306480
4062  5.16187 2.43447 1.290050
4063  5.04054 2.63783 1.499380
4064  4.76063 2.99594 1.190810
4065  4.86819 3.22996 1.581210
4066  5.06233 2.66640 1.393230
4067  4.92943 2.61109 1.181060
4068  4.82051 3.14140 1.393300
4069  4.53559 3.45942 1.399480
4070  4.86052 2.71287 1.151930
4071  4.98968 2.53473 1.202570
4072  4.86568 2.84438 1.247720
4073  4.81389 2.82299 1.200990
4074  5.08249 2.94725 1.253370
4075  4.61003 3.13926 1.270710
4076  4.97217 2.85614 1.233480
4077  5.02818 3.02591 1.181730
4078  4.65825 3.23443 1.361930
4079  4.96375 2.67542 1.153790
4080  4.97090 2.57031 1.266870
4081  5.05803 2.85375 1.369080
4082  4.64783 3.03224 1.351000
4083  4.81883 3.06241 1.094460
4084  5.33591 2.58568 1.254990
4085  4.82378 2.96861 1.240360
4086  4.90174 3.06562 1.248290
4087  4.71808 2.93775 1.301370
4088  5.04257 2.77546 1.131190
4089  4.87387 3.04827 1.141430
4090  4.81192 2.94028 1.239240
4091  4.92296 2.61860 1.128790
4092  5.38218 2.53851 1.403810
4093  5.04443 2.55551 1.284940
4094  5.00482 2.80099 1.322670
4095  4.70227 2.93028 1.401650
4096  4.97913 2.71121 1.172480
4097  4.74484 2.63140 1.403920
4098  4.86549 3.17112 1.240410
4099  4.67064 3.14978 1.227670
4100  4.88951 3.10150 1.210190
4101  4.72732 3.17887 1.267670
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4502  4.89666 2.58006 1.220340
4503  4.94109 2.81650 1.193040
4504  4.53572 3.28556 1.285000
4505  4.89104 2.92922 1.222750
4506  4.60924 3.19824 1.113720
4507  4.98131 2.95986 1.084030
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4509  5.04288 2.84384 1.187150
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4527  4.81680 3.01121 1.277690
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4537  4.97483 3.06182 1.271440
4538  4.80571 2.91438 1.194230
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4543  4.79211 2.82154 1.386090
4544  4.85446 2.89471 1.175260
4545  5.06783 2.83696 1.253040
4546  5.03382 2.91441 1.354580
4547  4.74784 3.00561 1.296910
4548  4.72671 2.91832 1.172610
4549  4.85068 2.82249 1.133300
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4551  5.02274 2.67293 1.234770
4552  5.19029 2.59217 1.353420
4553  4.73161 2.86859 1.265440
4554  5.08682 2.56576 1.434950
4555  4.93447 3.00276 1.256980
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4757  5.06168 2.60644 1.096260
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4802  4.79698 3.12310 1.133970
4803  4.92213 2.89117 1.103610
4804  4.94638 2.45532 1.270130
4805  5.09008 2.73420 1.170130
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4807  5.01696 2.70797 1.241810
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4951  4.85367 3.06194 1.293340
4952  5.04648 2.57533 1.227220
4953  5.16242 2.77207 1.242180
4954  4.70764 2.84805 1.185000
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4956  5.19967 2.42763 1.115530
4957  4.72501 3.08307 1.169140
4958  5.04602 2.91532 1.358230
4959  5.05910 2.89728 1.267760
4960  4.90961 2.76386 1.312220
4961  4.89112 2.98133 1.290000
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4963  5.08980 2.78962 1.202240
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4965  4.82182 2.86172 1.299750
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4972  4.78061 2.79811 1.366170
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4977  5.20821 2.85750 1.258790
4978  5.34908 2.19708 1.313350
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4984  4.91593 2.66803 1.151420
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4991  5.05690 2.91123 1.196540
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4993  4.94368 2.80832 1.154240
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5000  4.95452 2.81593 1.199550
5001  5.07073 2.71527 1.188220
5002  5.05095 2.41158 1.231010
5003  5.09117 2.68098 1.198780
5004  4.95070 2.73015 1.199020
5005  4.83655 3.01371 1.310880
5006  5.10024 2.54821 1.141360
5007  4.43454 3.14285 1.235600
5008  4.80867 3.14650 1.202530
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5410  4.69686 3.07287 1.329300
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5412  4.83479 3.03277 1.121730
5413  4.88561 2.68459 1.292140
5414  5.13072 2.59915 1.223830
5415  4.77538 3.28475 1.346250
5416  4.74268 3.26992 1.083770
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5424  4.85394 3.23034 1.333610
5425  4.70278 3.09418 1.189590
5426  4.73627 3.40870 1.341810
5427  4.63235 3.38840 1.313050
5428  4.78023 2.87328 1.276240
5429  4.71616 2.94891 1.312930
5430  4.68323 3.36831 1.411630
5431  4.69726 2.95171 1.261620
5432  4.93747 2.99407 1.302040
5433  4.75308 2.66708 1.160010
5434  5.10966 2.72186 1.199180
5435  4.92153 3.07610 1.238170
5436  4.83887 2.98584 1.232060
5437  4.69341 2.85735 1.201170
5438  4.99450 2.70588 1.323480
5439  5.13515 2.57632 1.118490
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5445  4.92717 2.67715 1.126650
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5448  4.85899 3.09457 1.133620
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5450  4.88517 2.85300 1.318600
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5452  5.25540 2.77876 1.367260
5453  4.81811 3.14869 1.316760
5454  4.86647 3.07300 1.155250
5455  5.09927 2.41731 1.319450
5456  4.72983 3.10165 1.199000
5457  4.82046 2.84143 1.195470
5458  4.86980 3.32864 1.245070
5459  5.00148 2.77996 1.243310
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6360  4.62568 3.30859 1.139130
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7230  4.98434 2.73454 1.193530
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7260  5.00082 2.39172 1.291270
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7262  4.80361 3.09987 1.213950
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7264  5.08951 2.75810 1.255070
7265  4.62562 3.27354 1.250660
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7267  4.84780 2.97221 1.309260
7268  4.86387 2.79070 1.157210
7269  5.05512 2.36619 1.353000
7270  4.93636 2.60496 1.305050
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7272  4.70774 2.70535 1.280180
7273  4.77484 3.28972 1.206970
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7280  5.02789 2.92797 1.426350
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7364  4.93340 2.78911 1.300360
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7429  5.07640 2.66430 1.165240
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7431  4.97007 2.71767 1.318270
7432  4.62198 3.20078 1.363280
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7434  4.98989 3.02953 1.533490
7435  5.15480 2.41102 1.223360
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7438  4.79246 2.93814 1.241010
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7502  4.99908 2.79080 1.307040
7503  4.75026 2.73011 1.153190
7504  5.02077 2.67187 1.196750
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7506  4.89033 3.02567 1.238210
7507  5.07712 2.68641 1.514020
7508  4.97430 2.50183 1.387750
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7680  4.96400 2.62182 1.113100
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7683  4.86395 3.03893 1.322600
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7703  4.91921 2.81734 1.164690
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7705  4.49531 3.34073 1.330300
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7710  4.92708 3.20549 1.336920
7711  4.89129 2.71566 1.167480
7712  4.94756 2.93491 1.238190
7713  4.98775 2.80122 1.200740
7714  4.82595 3.09450 1.198010
7715  4.88838 3.26260 1.268240
7716  5.22649 2.45457 1.313430
7717  5.00864 2.87768 1.081880
7718  4.86607 3.12509 1.139170
7719  4.89142 3.25918 1.422610
7720  4.99451 2.81434 1.266790
7721  5.12163 2.92999 1.190930
7722  4.61698 3.13630 1.240700
7723  5.15612 2.58785 1.143660
7724  4.91511 2.83110 1.137580
7725  4.66709 3.25463 1.254130
7726  5.25873 2.56795 1.152370
7727  5.29703 2.33604 1.287310
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8600  4.81010 3.06317 1.170730
8601  5.01984 2.43360 1.267410
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8610  4.97279 2.79951 1.289870
8611  4.69034 3.37058 1.149110
8612  4.92812 2.78669 1.118040
8613  4.62193 2.97859 1.231060
8614  4.71515 3.27763 1.415140
8615  4.89020 3.20133 1.134810
8616  4.72637 3.06570 1.337970
8617  4.98841 2.69503 1.210070
8618  5.01687 2.98499 1.319180
8619  5.19598 2.37667 1.180310
8620  5.43348 2.44847 1.302450
8621  4.97100 2.69920 1.277500
8622  5.13308 2.40120 1.269160
8623  4.92934 2.98253 1.159370
8624  4.88238 3.00858 1.155110
8625  4.94545 2.76575 1.249360
8626  4.97311 2.88633 1.222590
8627  4.64464 3.27392 1.143090
8628  4.90068 2.64852 1.339020
8629  4.98148 2.60173 1.204790
8630  4.85552 2.80781 1.213100
8631  5.12260 2.63017 1.266380
8632  4.93635 2.77109 1.191650
8633  5.01676 2.46211 1.339940
8634  4.86400 2.75353 1.262190
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8636  5.05992 2.66319 1.213140
8637  4.71317 3.18071 1.104660
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8640  4.83135 3.11388 1.233120
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8712  4.77148 3.10380 1.185300
8713  4.85329 3.00711 1.195460
8714  4.96361 3.05004 1.130730
8715  5.15313 2.49387 1.219180
8716  4.83422 3.09057 1.260510
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8727  4.95061 3.00471 1.351440
8728  4.96540 2.50279 1.302830
8729  5.00643 2.73204 1.238710
8730  5.03059 2.47675 1.294800
8731  4.94601 2.60976 1.174480
8732  4.49463 3.10881 1.269110
8733  5.01759 2.66826 1.225150
8734  4.58239 3.00035 1.343000
8735  4.79793 3.00429 1.342980
8736  4.75320 3.13718 1.228150
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8738  5.30664 2.39845 1.279330
8739  4.79534 2.97722 1.209160
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8760  4.71764 3.11271 1.157040
8761  5.27473 2.65276 1.354350
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8765  4.70590 3.27086 1.226700
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8767  4.91362 2.83918 1.389070
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8769  4.84552 2.83618 1.200260
8770  5.03164 3.12757 1.280730
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8775  4.88782 2.83110 1.130700
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8790  4.62799 3.25134 1.396170
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8794  4.44668 3.59827 1.287630
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8813  4.95840 2.65535 1.219330
8814  4.88615 2.62351 1.161000
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8817  5.15109 2.72552 1.338180
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8820  4.92395 3.19269 1.270530
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8826  4.68766 2.98564 1.212100
8827  4.68461 2.96488 1.168560
8828  5.31513 2.63549 1.219340
8829  4.86022 3.02537 1.238920
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8831  4.60406 3.44400 1.447090
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8833  5.04437 2.76834 1.381220
8834  4.81313 3.05341 1.172510
8835  4.79201 3.15620 1.210050
8836  4.92895 3.09573 1.244720
8837  4.87015 2.72826 1.397030
8838  4.62189 3.24278 1.199950
8839  5.16940 2.46898 1.263980
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8846  5.24184 2.82561 1.189550
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8849  4.90294 3.08822 1.111130
8850  4.84108 2.96694 1.249390
8851  5.05490 2.53977 1.174420
8852  4.89342 2.50482 1.393930
8853  5.06710 2.55802 1.228640
8854  4.98648 2.82253 1.141050
8855  4.85108 2.64952 1.275730
8856  4.49968 3.57810 1.273820
8857  4.84362 2.72662 1.211180
8858  5.04960 2.82873 1.274820
8859  4.81799 3.28747 1.368900
8860  5.13031 2.28753 1.209060
8861  4.87910 3.12328 1.088960
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8863  4.95819 2.76090 1.486750
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8865  4.93933 2.99541 1.177100
8866  4.91286 3.03198 1.225210
8867  5.01265 2.87626 1.315520
8868  4.73230 3.22692 1.194110
8869  4.72803 3.05964 1.208120
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8904  4.75614 2.90035 1.315540
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8910  4.86719 2.95340 1.259200
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8913  4.67057 3.23347 1.305720
8914  5.08982 2.90409 1.327560
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8918  4.94250 2.60735 1.213920
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8923  4.96691 2.24390 1.363640
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8926  4.86347 2.96634 1.384000
8927  4.97361 2.99414 1.116070
8928  5.34485 2.23384 1.342250
8929  5.19980 2.57405 1.293420
8930  4.96989 2.89855 1.232030
8931  5.13937 2.72285 1.161110
8932  4.66054 3.28863 1.212200
8933  4.78088 3.23946 1.210030
8934  5.16286 2.47292 1.256360
8935  4.88159 2.65164 1.260970
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8939  5.04210 2.72973 1.116350
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9027  5.12721 2.78506 1.161960
9028  4.95771 2.94402 1.129670
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9030  4.43436 3.67479 1.488080
9031  4.94727 3.13704 1.448000
9032  4.90699 2.95800 1.141640
9033  4.97571 2.63622 1.069440
9034  5.08257 2.69717 1.342760
9035  4.99463 2.74217 1.305910
9036  5.25745 2.48970 1.481800
9037  4.93828 3.12792 1.327190
9038  4.76718 2.98071 1.184910
9039  4.89860 2.98917 1.327250
9040  4.92426 2.83883 1.230030
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9057  4.86205 2.82460 1.222030
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9059  4.98382 2.76248 1.261690
9060  5.06108 2.88253 1.430840
9061  4.64705 3.06640 1.292890
9062  5.01554 2.92952 1.179890
9063  4.99408 2.88573 1.295050
9064  5.13446 2.68585 1.128270
9065  4.74139 2.99311 1.338760
9066  5.03870 2.51411 1.285520
9067  5.03062 2.73544 1.214770
9068  5.10362 2.68906 1.314300
9069  5.05992 2.53197 1.328080
9070  4.67099 3.09806 1.275470
9071  4.88799 2.86771 1.338980
9072  5.12178 2.74977 1.336240
9073  4.94052 2.86770 1.320570
9074  4.98351 2.87737 1.376450
9075  4.91678 2.99330 1.252590
9076  5.02038 2.95226 1.166400
9077  4.89489 2.90693 1.200730
9078  4.90675 2.82119 1.159460
9079  5.16809 2.47470 1.386200
9080  5.08063 2.42870 1.219910
9081  4.76648 2.84111 1.204530
9082  4.66895 3.17761 1.159350
9083  4.75053 3.12886 1.264680
9084  4.95422 2.73174 1.177700
9085  4.95964 2.90294 1.218340
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9537  4.94541 2.96312 1.055760
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9991  4.95172 3.01510 1.421540
9992  4.87127 3.00536 1.348980
9993  5.08729 2.51983 1.163570
9994  5.26011 2.61447 1.145540
9995  4.88565 2.64869 1.358230
9996  4.74631 3.06858 1.272010
9997  4.78326 3.17788 1.356780
9998  4.69706 3.17782 1.073870
9999  4.78578 2.74896 1.410020
10000 4.84193 3.04107 1.077230

Distributions of posterior parameter estimates

dens(post)

Estimate for group 1

Combine the Group 0 estimate with b1 for each sample:

post <- post |> 
  mutate(Group1 = Group0 + b1)

head(post)

   Group0      b1   sigma  Group1
1 5.02212 2.54851 1.19435 7.57063
2 5.18027 2.59637 1.29033 7.77664
3 4.87321 2.97078 1.24254 7.84399
4 5.02465 2.85688 1.27407 7.88153
5 4.88022 2.70676 1.27963 7.58698
6 4.88433 2.80594 1.26673 7.69027

Plotting posterior distributions

post |> 
  dplyr::select(Group0, Group1) |> 
  pivot_longer(cols = everything(),
               names_to = "Group",
               values_to = "Estimate") |> 
  ggplot(aes(Estimate, fill = Group)) +
  geom_density()

Plotting posterior distributions

HPDIs

89% highest posterior density intervals

HPDI(post$Group0)

  |0.89   0.89| 
4.62897 5.18668

HPDI(post$Group1)

  |0.89   0.89| 
7.49595 8.06625

Modeling categorical predictors by group

Use 1, 2, 3, etc. for groups
Use [ ] notation in the model statement

\[y \sim Normal(\mu, \sigma)\]

\[\mu = b[Group12]\]

b[Group12] tells stan to estimate each level of Group12 separately

Prior prediction

Mean for groups probably $0 \pm 10$
Conservative to assume both are drawn from the same distribution

set.seed(742999)

sim_model <- function(ii) {
  Group0 <- rnorm(n, 0, 10)
  Group1 <- rnorm(n, 0, 10)
  tibble(rep = ii,
         Group0 = mean(Group0),
         Group1 = mean(Group1))
}

sims <- purrr::map(.x = 1:500,
            .f = sim_model) |> 
  list_rbind()
print(sims, n = 200)

Prior prediction

# A tibble: 500 × 3
      rep   Group0   Group1
    <int>    <dbl>    <dbl>
  1     1 -0.668   -1.66   
  2     2 -1.20    -1.52   
  3     3  2.67     0.170  
  4     4  1.71     1.91   
  5     5 -0.221   -1.36   
  6     6  1.99     0.645  
  7     7 -2.30     0.300  
  8     8  1.05     1.32   
  9     9  1.54     1.48   
 10    10  0.558   -2.31   
 11    11 -1.56    -0.288  
 12    12 -0.900   -1.35   
 13    13  0.942    0.236  
 14    14 -1.72    -3.43   
 15    15  0.0290  -2.35   
 16    16 -0.739   -0.214  
 17    17  2.09    -1.67   
 18    18  0.903   -2.55   
 19    19  0.572   -1.18   
 20    20  3.53    -0.638  
 21    21 -0.152    2.39   
 22    22 -1.12    -0.629  
 23    23  1.01    -1.51   
 24    24 -0.564    0.253  
 25    25  1.26    -0.201  
 26    26 -0.653    1.62   
 27    27 -1.00     0.980  
 28    28 -0.129   -0.465  
 29    29  0.981    0.726  
 30    30 -0.807    1.23   
 31    31  0.723   -1.33   
 32    32  0.261    1.02   
 33    33  1.54     0.722  
 34    34 -0.489   -2.75   
 35    35  0.832    0.00489
 36    36 -0.358    2.91   
 37    37 -2.61    -0.930  
 38    38 -0.688    0.596  
 39    39 -1.01    -1.79   
 40    40 -1.29     0.799  
 41    41 -1.23    -1.28   
 42    42 -0.0603  -1.25   
 43    43 -0.128   -1.43   
 44    44 -0.663   -0.936  
 45    45  2.23    -0.795  
 46    46  1.89    -1.11   
 47    47 -0.315    2.23   
 48    48 -1.37     0.405  
 49    49 -0.524   -0.144  
 50    50 -0.207   -1.23   
 51    51 -1.09     2.74   
 52    52  0.164   -2.27   
 53    53 -0.129    1.03   
 54    54 -1.78    -0.0924 
 55    55  0.272   -1.07   
 56    56 -1.90    -0.0318 
 57    57  0.230    0.547  
 58    58 -0.590   -1.40   
 59    59  2.03    -2.50   
 60    60  0.196    1.96   
 61    61 -0.0822  -1.25   
 62    62 -1.42     2.31   
 63    63  0.371    0.150  
 64    64  0.723   -1.04   
 65    65  0.271   -2.88   
 66    66  1.50    -0.561  
 67    67 -0.558   -2.03   
 68    68 -0.879    3.05   
 69    69 -2.11    -0.284  
 70    70  1.36     0.0817 
 71    71 -0.158    0.0984 
 72    72 -2.52     1.51   
 73    73 -0.919    0.192  
 74    74  1.60     0.660  
 75    75 -2.62     0.824  
 76    76  0.331   -1.76   
 77    77  1.34    -1.27   
 78    78  2.82     0.982  
 79    79  1.04     2.09   
 80    80  0.206    1.68   
 81    81 -1.26     2.16   
 82    82  1.59    -0.790  
 83    83 -3.63    -2.99   
 84    84 -2.57     2.21   
 85    85 -0.905   -1.47   
 86    86  2.14    -0.218  
 87    87  1.58    -3.32   
 88    88 -0.457   -2.63   
 89    89 -1.25    -0.234  
 90    90  0.882    1.10   
 91    91  0.758    1.20   
 92    92  3.71    -1.04   
 93    93  0.417    2.60   
 94    94 -3.14     1.81   
 95    95  0.776   -0.906  
 96    96  0.594    2.14   
 97    97  0.738   -0.115  
 98    98  0.465   -2.92   
 99    99  2.13    -0.617  
100   100  1.07    -0.358  
101   101 -1.71     0.289  
102   102  0.990   -2.35   
103   103 -1.21     0.812  
104   104  1.90     0.623  
105   105  0.0506  -3.54   
106   106  0.119    0.749  
107   107 -1.02     0.961  
108   108 -2.72    -0.327  
109   109 -1.27    -1.42   
110   110  0.946   -2.74   
111   111  0.706    0.143  
112   112 -1.18    -0.0289 
113   113 -0.785    0.712  
114   114 -1.70     2.46   
115   115  1.88    -0.0644 
116   116  0.810    0.427  
117   117  1.42     0.281  
118   118  0.234    1.93   
119   119 -1.37     1.03   
120   120 -1.89     0.231  
121   121 -2.38     0.893  
122   122 -0.0144  -0.319  
123   123 -1.30     0.00456
124   124  1.06    -1.32   
125   125  0.106   -0.657  
126   126  2.10     0.459  
127   127  1.05    -0.879  
128   128  0.159   -2.05   
129   129  0.701    2.23   
130   130 -1.53    -2.25   
131   131  1.27    -2.29   
132   132 -0.942    0.683  
133   133  1.72    -0.380  
134   134  2.04     0.686  
135   135 -0.569   -0.601  
136   136 -0.0966  -0.833  
137   137 -1.44    -0.831  
138   138  0.644    1.04   
139   139  1.32    -1.85   
140   140 -0.978    0.234  
141   141  0.448   -0.740  
142   142 -1.20    -0.240  
143   143  0.179    1.31   
144   144  0.435    0.115  
145   145 -0.372    1.19   
146   146 -1.04    -2.54   
147   147  0.396    1.10   
148   148  1.26     2.20   
149   149 -3.31     1.89   
150   150 -1.09    -1.02   
151   151  3.03    -0.383  
152   152 -1.28    -1.03   
153   153 -1.99     1.14   
154   154  1.08     0.571  
155   155 -0.718   -2.01   
156   156  0.643    1.93   
157   157  1.53     1.44   
158   158 -0.0262   0.804  
159   159  3.31    -0.869  
160   160  0.436   -0.204  
161   161  0.308    1.18   
162   162  1.25     0.336  
163   163 -0.00656  1.52   
164   164  0.788   -0.280  
165   165  0.228   -1.48   
166   166 -1.05     0.274  
167   167  0.0459  -1.32   
168   168 -3.11     0.0518 
169   169 -0.963   -1.52   
170   170  0.667   -0.227  
171   171  2.01     1.32   
172   172  1.90     2.00   
173   173  1.18     2.80   
174   174 -0.356   -1.78   
175   175 -0.798    0.755  
176   176  0.221   -0.905  
177   177  1.46     0.532  
178   178 -0.837   -1.32   
179   179 -0.799    1.38   
180   180  0.792    0.446  
181   181 -1.01     0.733  
182   182 -0.261   -0.970  
183   183  0.660   -1.97   
184   184 -0.208   -0.404  
185   185  1.00     3.73   
186   186 -0.681   -1.30   
187   187 -1.31    -2.08   
188   188 -2.65    -0.356  
189   189  0.262    0.711  
190   190  0.246   -0.673  
191   191 -0.886   -1.36   
192   192 -1.39    -0.0805 
193   193  2.12    -1.22   
194   194 -1.64     0.131  
195   195 -1.88     0.298  
196   196  0.641   -1.92   
197   197 -0.598    0.155  
198   198  0.898   -1.16   
199   199 -1.68    -1.05   
200   200 -1.09    -3.40   
# … with 300 more rows

Plotting the prior predictions

Are the predicted outcomes on the right scale?

sims |> 
  dplyr::select(-rep) |> 
  pivot_longer(cols = everything(),
               values_to = "Mean",
               names_to = "Group") |>
  ggplot(aes(x = Group, y = Mean)) +
  geom_point(position = position_jitter(width = 0.25))

Plotting the prior predictions

Fitting the model

fm <- ulam(
  alist(
    y ~ dnorm(mu, sigma),
    mu <- b[Group12],
    b[Group12] ~ dnorm(0, 10),
    sigma ~ dhalfnorm(0, 3)
  ),
  data = D,
  chains = 4,
  iter = 5e3
)

Inspecting the chains: `traceplot`

traceplot(fm)

Inspecting the chains: Rank histogram

trankplot(fm)

Inspecting the output

Note depth = 2 to return the levels of b

precis(fm, depth = 2)

          mean         sd     5.5%    94.5%     n_eff     Rhat4
b[1]  4.908901 0.17380793 4.632475 5.182223 10545.118 1.0001915
b[2]  7.782463 0.17710677 7.496329 8.067577  9164.483 1.0002584
sigma 1.241821 0.08845105 1.109309 1.388342  9595.490 0.9997953

Posterior

Keep only the b estimates using $b:

post <- extract.samples(fm)$b |> 
  as.data.frame() |> 
  rename(`Group 0` = V1,
         `Group 1` = V2)

post |> 
  pivot_longer(cols = everything(),
               names_to = "Group",
               values_to = "Estimate") |> 
  ggplot(aes(Estimate, fill = Group)) +
  geom_density()

Bivariate models and regularizing priors